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Distribution of zeros of entire functions with a subharmonic majorant B. N. Khabibullin Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, Russia
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    § 1. Introduction1.1. Motivations, origins and the statement of the main problemThe problem below is of interest not only as one of the key intrinsic questions in the theory of growth of entire functions, that is, holomorphic functions on the complex plane $\mathbb{C}$ [1], [2]. It also frequently arises in approximation, representation by series, interpolation, spectral analysis and synthesis, the theory of algebraic structures in classes of functions, the spectral theory of operators, analytic number theory and so on. We present first the statement of the main problem, deferring the rigourous consistent definitions to what follows. Main Problem. Let $Z$ be a distribution of points and $M$ be a subharmonic function on $\mathbb{C}$, and assume that there exists an entire function $f\neq 0$ vanishing on $Z$ such that $\log|f|\leqslant M$ on $\mathbb{C}$. What are the relationships between $Z$ and the Riesz mass distribution $\varDelta_M$ of $M$? Once a statement describing such relationships is established, the converse to the opposite statement has the form of a uniqueness theorem. One aim of this paper is to deduce such uniqueness theorems. Using integral transformations we can often realize the topological dual of a space of functions defined on subsets of $\mathbb{C}$ as a weighted space of entire functions. In this case uniqueness theorems of this type produce results on the completeness of various systems of functions in the original function space. We discuss elsewhere their applications to the completeness of exponential and more general parametrized systems of entire functions in spaces of holomorphic functions, because in this case the main results on completeness can be stated in a pronouncedly geometric form [3]–[5]; however, this requires a considerable preliminary work, which is also one of our aims in this paper. The fact that in the main problem we deal specifically with subharmonic majorizing functions $M \geqslant \log|f|$ is not a serious constraint, because for an arbitrary majorant $M \geqslant \log|f|$ we can always find a subharmonic function between $\log |f|$ and $M$, defined as the greatest subharmonic minorant of $M$. The comparison of the distribution of points $Z$ with the Riesz mass distribution $\varDelta_M$ of $M$, rather than with $M$ itself, is quite natural, as justified in § 5.3 below. Nevertheless, historically, authors mostly considered the variation of the main problem in which they looked for relationships between $Z$ and the majorant $M$ itself. Moreover, conditions of the type $\log|f|\leqslant M$ had a must more relaxed form, in terms of a system of majorizing weights $M$ with considerable gaps between them. However, as a rule and, in particular, in applications of the main problem we see more-or-less concrete majorizing functions $M\geqslant \log|f|$ or some special weight classes of such functions. In these cases one can distinguish an explicit link between the subharmonic function $M$ and the corresponding distribution of Riesz masses $\varDelta_M$ in one form or another. One of the simplest ‘radial’ versions of such a link is the classical formula of Poisson–Jensen–Nevanlinna–Privalov ([6], [7], Ch. II, § 2, and [8], § 3.7). It connects the counting radial function $\varDelta_M^{\mathfrak{r}}(t)=\varDelta_M \bigl(\overline D(t)\bigr)$ of the Riesz mass distribution $\varDelta_M$ over the closures $\overline D(r)$ of open discs $D(r)$ of radius $r>0$ with centre at zero and the integral means of $M$ on the circles $\partial \overline D(r)$ of radius $r$ with centre at zero, which we denote in what follows by
$$
\begin{equation}
M^{\circ r}:=\frac{1}{2\pi}\int_0^{2\pi}M(re^{i\theta})\,\mathrm{d} \theta\geqslant M(0) \quad\text{for } r\geqslant 0.
\end{equation}
\tag{1.1}
$$
For a subharmonic function $M\neq -\infty$ in $\mathbb{C}$ the difference version of the Poisson–Jensen–Nevanlinna–Privalov formula ([6] or [9], § 2) is more important. Using the notation (1.1) it can be written as
$$
\begin{equation}
\int_r^R\frac{\varDelta_M^{\mathfrak{r}}(t)}{t}\,\mathrm{d} t \overset{(1.1)}{=}M^{\circ R}-M^{\circ r} \quad\text{for all } 0<r<R<+\infty.
\end{equation}
\tag{1.2}
$$
If an entire function $f\neq 0$ vanishes on $Z$ (and can have other zeros away from $Z$) and, in addition, $\log |f| \leqslant M$ on $\mathbb{C}$, and if $Z^{\mathfrak{r}}(t)$ is the number of points in $Z$ occurring in $\overline D(t)$, then
$$
\begin{equation*}
\begin{aligned} \, \int_r^R\frac{Z^{\mathfrak{r}}(t)}{t}\,\mathrm{d} t &\overset{(1.2)}{\leqslant}(\log|f|)^{\circ R}-(\log|f|)^{\circ r}\leqslant M^{\circ R}-(\log|f|)^{\circ r} \\ &\overset{(1.2)}{=}\int_r^R\frac{\varDelta_M^{\mathfrak{r}}(t)}{t}\,\mathrm{d} t+M^{\circ r}-(\log|f|)^{\circ r} \quad\text{for all } 0< r<R<+\infty. \end{aligned}
\end{equation*}
\notag
$$
Hence under the assumption $f(0)\neq 0$ (which means that $Z^{\mathfrak{r}}(0)=0$), since $(\log|f|)^{\circ r}\overset{(1.1)}{\geqslant}\log|f(0)|$, we obtain an elementary but rather interesting inequality
$$
\begin{equation}
\int_r^R(Z^{\mathfrak{r}}(t)-\varDelta_M^{\mathfrak{r}}(t))\frac{1}{t}\,\mathrm{d} t\leqslant M^{\circ r}-\log|f(0)| \quad\text{for all } 0< r<R<+\infty.
\end{equation}
\tag{1.3}
$$
Not only the constraints imposed on the left-hand side are important here for $R$ approaching $+\infty$, but also the fact that $r$ takes values between $0$ and $R$, which was generally not mentioned before. This is the form we bear in mind in our main results. The first important advancement in our results comparing $Z$ and $\varDelta_M$ and extending (1.3) is the replacement of the radial counting functions $Z^{\mathfrak{r}}$ and $\varDelta_M^{\mathfrak{r}}$ in (1.3) by the radial-angular counting functions $Z^{\mathfrak{ra}(s)}$ with weight $s$ for the arguments of points in $Z$, and the corresponding transition to the radial-angular counting functions $\varDelta_M^{\mathfrak{ra}(s)}$ also for the Riesz mass distributions $\varDelta_M$. Thus, in our results here we take account, in fact, not only of the distribution of the moduli of points in $Z$ but also of their distribution in the argument. As the weight $s$ with respect to the argument we take arbitrary positive $p$-trigonometrically convex functions for all possible values of $p\geqslant 0$ [1], [2], [10], [11]. Second, in the integrand in (1.3) we can use functional coefficients from a very wide class of functions on the interval $(r,R)$ in place of the particular function $1/t$ in front of the differential $\,\mathrm{d} t$. This class of functions, which we call $p$-power convex in what follows, is apparently used for the first time in investigations of the distributions of zeros of entire functions (by contrast to the class of $p$-trigonometrically convex functions, which was often used in the theory of growth of entire functions [1], [2], [10]–[18]. As concerns $ p$-power convex functions themselves, similarly to $p$-trigonometrically convex functions, they are special cases of generalized convex functions in the sense of Valiron and Beckenbach [19]–[21], which were briefly reviewed in the monograph of Roberts and Varberg (see [22], Ch. VIII, § 84). In this paper we investigate $ p$-power convex functions only within the framework of their usefulness for the main results. The third important aspect is that inequalities for the distributions of zeros are uniform over the classes of $p$-trigonometric and $ p$-power convex functions, provided that they are normalized and the value $f(0)\neq 0$ of the function $f$ vanishing on $Z$ is fixed. Hence we can apply our univariate inequalities to entire functions of several complex variables and the distributions of their zeros in $n$-dimensional complex space $\mathbb{C}^n$. We are planning to do this in subsequent papers. Finally, a few of our criteria obtained in 2019–2021, in joint papers with F. Khabibullin (see [23], Theorem 2 and Corollaries 1 and 2), for subharmonic or $\delta$-subharmonic majorizing functions $M \geqslant \log|f|$ (see [24], Theorem 3) solve fully the main problem under a very mild single condition: there exists $P\geqslant 1$ such that
$$
\begin{equation}
\sup_{z\in \mathbb{C}} \biggl(\frac{1}{2\pi}\int_0^{2\pi}M\biggl(z+\frac{e^{i\theta}}{(1+|z|)^P}\biggr) \,\mathrm{d} \theta-M(z)\biggr)<+\infty.
\end{equation}
\tag{1.4}
$$
These criteria in [23] and [24] use various test classes of Jensen potentials, generalized Greens functions with pole at zero or logarithmic potentials of analytic discs, or measures with a stringent condition of logarithmic (semi)normalization at zero (or at infinity, in the case of an inversion). All this makes the use of the above criteria more difficult. Using $p$-trigonometrically or $ p$-power convex functions for tests we can abandon the framework of this logarithmic (semi)normalization and condition (1.4), although at this point only in what concerns new necessary relations between $Z$ and $M$. Perhaps for some classes of majorizing functions our new, more geometric and more easily comprehensible necessary relations between $Z$ and $\varDelta_M$ are also sufficient. Almost all results preceding 2012 on the main problem can be found in our monographic survey [17]. The key ones are formulated in § 1.3 as Theorems 1.1 and 1.2. These two uniqueness theorems are very special cases of our main results in this paper, which we illustrate by Theorem 2.2 in § 2.3. In § 1.3 we also state as Theorem 1.3 a version of the criterion that solves fully the main problem in the framework of condition (1.4). Our main results on the distribution of zeros of entire functions, Theorems 2.1 and 2.3 and some of their corollaries, are presented in § 2. We prove them in § 5 using our main theorem (Theorem 5.1), after some preparatory work in §§ 3 and 4. The general ‘subharmonic’ version of the approach to the main problem is stated as the main theorem (Theorem 5.1) in § 5 and proved there not only for functions on the plane, but also for functions on some disc, for instance, the unit one. We are also planning to look closer in the future at applications of the version of the main theorem for functions on the unit disc that develops our joint results with F. Khabibullin (see [18], the main and uniqueness theorems). The reader can address the next subsection (§ 1.2) as needed, when questions on definitions, notation and terminology arise. We present some of these with a ‘margin’, not only for $\mathbb{C}$, but also for domains in $\mathbb{C}$. This is also valid for general auxiliary integral inequalities for differences of subharmonic functions in § 3. 1.2. Some notation, definitions and conventions1.2.1. Number sets, intervals and discsWe denote the empty set by $\varnothing$, $\mathbb N:=\{1,2, \dots\}$ is the set of natural numbers, ${\mathbb{N}_0:=\{0\}\cup \mathbb{N}=\{0,1, 2, \dots\}}$, and $\overline{\mathbb{N}}_0:=\mathbb{N}_0\cup \{+\infty\}$ is the upper order completion of the set $\mathbb{N}_0$ with the standard order relation $\leqslant$ and supremum $+\infty:=\sup \mathbb{N}_0\notin \mathbb{N}_0$; we have the inequalities $n\leqslant +\infty$ for all $n\in \overline{\mathbb{N}}_0$. We regard the set of real numbers $\mathbb{R}$ with order relation $\leqslant$, in particular, as the real axis in $\mathbb{C}$ with Euclidean norm given by the modulus $|\cdot|$ and positive half-axis $\mathbb{R}^+:=\{x\in \mathbb{R}\mid| x\geqslant 0\}$. The order completion of $\mathbb{R}$ by the supremum $+\infty :=\sup \mathbb{R}=\inf\varnothing \notin \mathbb{R}$ and infimum $-\infty :=\inf \mathbb{R}=\sup\varnothing\notin \mathbb{R}$ is the extended real axis $\overline{\mathbb{R}}:= \mathbb{R}\cup \{ \pm \infty\}$ with natural operations and exceptions from these (see [9], § 2, formula (2.8), and with the convention $0\cdot \infty:=0=: \infty \cdot 0$, unless otherwise stated. A quantity $x\in \overline{\mathbb{R}}$ is said to be positive if $x\in \overline{\mathbb{R}}^+$, strictly positive if $0\neq x\in \overline{\mathbb{R}}^+$, and it is negative or strictly negative if the opposite quantity $-x$ is positive or strictly positive, respectively. Furthermore, $x^+:=\sup \{0,x\}\in \overline{\mathbb{R}}^+$ is the positive part of the quantity $x\in \overline{\mathbb{R}}$, and $x^-:=(-x)^+\in \overline{\mathbb{R}}^+$ is its negative part. A segment in $\overline{\mathbb{R}}$ with left-hand endpoint $a\in \overline{\mathbb{R}}$ and right-hand endpoint $b\in \overline{\mathbb{R}}$ is the set $[a,b]:=\{x\in \overline{\mathbb{R}}\mid a\leqslant x\leqslant b\}$. A subset $I$ of $ \overline{\mathbb{R}}$ is convex if for each pair $a\in I$ and $b\in I$ it contains the segment $[a,b]$. We call a convex subset of $\overline{\mathbb{R}}$ an interval in $\overline{\mathbb{R}}$ with left-hand endpoint $\inf I\in \overline{\mathbb{R}}$ and right-hand endpoint $\sup I$. The set of intervals in $\overline{\mathbb{R}}$ consists of the segments, intervals of the form $(a,b]:=[a,b]\setminus \{a\}$ or $[a,b):=[a,b]\setminus \{b\}$ and open intervals $(a,b):=[a,b)\setminus \{a\}$, $(a,+\infty]$ and $[-\infty, b)$ in $\overline{\mathbb{R}}$. Open intervals form a base of open subsets of $\overline{\mathbb{R}}$. The notions of a convex and a connected set coincide in $\overline{\mathbb{R}}$. Each subset of $\overline{\mathbb{R}}$ is considered with the topology induced from $\overline{\mathbb{R}}$. We let $\overline D(r):=\{z\in \mathbb{C}\mid |z|\leqslant r\}$, $D(r):=\{z\in \mathbb{C}\mid |z|<r\}$ and $\partial \overline D(r):=\overline D(r)\setminus D(r)$ denote the closed and open discs and the circle with centre $0$ and radius $r\in \overline{\mathbb{R}}$. For $S\subset \mathbb{C}$ we denote by $\overline S$ and $\partial S$ the closure and boundary of $S$ in the extended complex plane $\mathbb{C}_{\infty}:=\mathbb{C}\cup \{\infty\}$ with a basis of open sets consisting of the open discs $z+D(r)$ for all $z\in \mathbb{C}$ and the complements to discs $\mathbb{C}_{\infty}\setminus \overline D(r)$ for all $r\in \mathbb{R}^+$. For instance, the inclusion $\overline S\subset \mathbb{C}$ means that $\overline S$ is a compact subset of $\mathbb{C}$. 1.2.2. FunctionsScalars $c\in \overline{\mathbb{R}}$ are often also regarded as the constant functions equal to $c$ identically. For instance, we can denote by $0$, apart from $0\in \overline{\mathbb{R}}$, also the zero function or the neutral element in the additive sense, and given a function $u$, we write, for instance, $u\neq -\infty$, to indicate that this function also takes values other than $-\infty$ in its domain of definition. For a more detailed description of a function $f\colon X\to Y$ we will often use the notation where on the right-hand side we indicate the rules for constructing $f(x)$. Some components of the notation can be dropped when they are clear from the context. Order relations for extended real functions $f\colon X\to \overline{\mathbb{R}}$ are treated pointwise. Furthermore, $f^+:=\sup\{f,0\}$ and $f^{-}:=(-f)^+$ are called the positive and negative parts of the function $f$. The function $f$ is positive (negative), which is denoted by $f\geqslant 0$ ($f\leqslant 0$), if $f=f^+$ ($f=-f^-$, respectively). A positive (negative) function $f$ is strictly positive, which is denoted by $f>0$ (strictly negative, which is denoted by $f<0$, respectively) if it does not take the value $0$.A function $f$ is bounded away from zero on $X$ if $\inf f(X)> 0$ or $\sup f(X)<0$. If $X\subset \overline{\mathbb{R}}$ and for all $x_1,x_2\in X$ it follows from $x_1<x_2$ that the nonstrict inequality $f(x_1)\leqslant f(x_2)$ (the strict inequality $f(x_1)< f(x_2)$) holds, then $f$ is an increasing (strictly increasing, respectively) function on $X$; $f$ is decreasing (strictly decreasing) on $X$ if the opposite function $-f$ is increasing (strictly increasing, respectively) there. Given a function $f$ on $X$ taking values in $\mathbb{C}$ or $\overline{\mathbb{R}}$, with modulus $|f|\colon x\underset{x\in X}{\longmapsto} |f(x)|$, set 1.2.3. Distributions of points and zerosLet $D\subset \mathbb{C}$ be a domain, that is, an open connected set. We call an arbitrary function $Z\colon D\to \overline{\mathbb{N}}_0$ a distribution of points on $D$ (see [17], §§ 0.1.2 and 0.1.3), with multiplicities $Z(z)\in \overline{\mathbb{N}}_0$ of the points $z\in D$ in $Z$. If $f$ is a holomorphic function on $D$, then we call the distribution of points
$$
\begin{equation}
\operatorname{Zero}_f\colon z\underset{z\in D}{\longmapsto} \sup\biggl\{p\in \mathbb{R}\biggm| \limsup_{z\neq w\to z}\frac{|f(w)|}{|w-z|^p}<+\infty \biggr\}\in \overline{\mathbb{N}}_0
\end{equation}
\tag{1.6}
$$
the distribution of zeros of the holomorphic function $f$ on $D$. According to a theorem of Weierstrass, either the distribution of zeros $\operatorname{Zero}_f$ has a finite multiplicity at each point $z\in D$ and $f\neq 0$, or $\operatorname{Zero}_f=+\infty$ on $D$ and $f=0$. A function $f$ vanishes on $Z$ if $\operatorname{Zero}_f\geqslant Z$ on $D$. We call a distribution of points $Z$ on $D$ a uniqueness distribution with respect to a function $M\colon D\to \overline{\mathbb{R}}$ on $D$ if two arbitrary holomorphic functions $f$ and $g$ on $D$ such that their difference $f-g$ vanishes on $Z$ coincide on $D$, provided that $\log|f|\leqslant M$ and $\log|g|\leqslant M$ on $D$. Otherwise we call $Z$ a nonuniqueness distribution with respect to $M$. In part 2 of Proposition 5.1 in § 5.3 we point out that $Z$ is a nonuniqueness distribution with respect to $M$ if and only if there exists a holomorphic function $f\neq 0$ on $D$ that vanishes on $Z$ and satisfies $\log|f|\leqslant M$ on $D$. We call such a function $f\neq 0$ a realization of the nonuniqueness distribution $Z$ with respect to $M$. For $z\in \mathbb{C}$ let $\arg z\subset \mathbb{R}$ denote the set of values of the angular arguments of $z$; $\arg 0:=\mathbb{R}$. Given a $2\pi$-periodic function $s\colon \mathbb{R}\to \mathbb{R}$, the quantities $s(\arg z)$ are well defined for all $z\in \mathbb{C}\setminus \{0\}$; for $z=0$ we set
$$
\begin{equation}
s(\arg 0):=\|s\|_\mathbb{R}{\overset{(1.5)}{=}}\sup_{\theta\in \mathbb{R}}\bigl|s(\theta)\bigr|.
\end{equation}
\tag{1.7}
$$
Given a positive function $s\geqslant 0$ the counting radial-angular function for the distribution of points $Z$ with weight $s$ in $\mathbb{C}$ is the positive increasing right-continuous function
$$
\begin{equation}
Z^{\mathfrak{ra}(s)}(t)\underset{ t\geqslant 0}{:=} \sum_{|z|\leqslant t}Z(z)s(\arg z) \in \overline{\mathbb{R}}^+
\end{equation}
\tag{1.8}
$$
on $\mathbb{R}^+$ ([13]–[18]). In particular, for $s= 1$ we have the usual counting radial function
$$
\begin{equation}
Z^{\mathfrak{r}}\colon t\underset{t\geqslant 0}{\longmapsto}Z^{\mathfrak{ra}(1)}(t)=\sum_{|z|\leqslant t}Z(z),
\end{equation}
\tag{1.9}
$$
which we used already in (1.3). By contrast to it, the counting radial-angular function (1.8) with weight $s$ not constant in the argument, is capable of taking into account quite delicately the distribution of points $Z$, not only as regards the radii of these points, but also as regards their arguments. In our paper we construct counting radially-angular functions (1.8) from $p$-trigonometrically convex weight functions $s$. 1.2.4. Distributions of masses and chargesAs concerns the theory of measure and integration, we use the terminology (but not always the notation) from the books [25] by Landkof (the introduction and § 1), [26] by Evans and Gariepi, and [27] by Ransford (appendix A). For example, an extended positive number function $\mu$ on the set of subsets of a set $X$ is called an (exterior) measure on $X$ if it is countably subadditive and $\mu(\varnothing )=0$. The concepts of Borel measure and regular measure on subsets of $\mathbb{R}$ or $\mathbb{C}$ are standard. A Radon measure on $\mathbb{R}$ or $\mathbb{C}$ is a Borel regular measure which is finite on compact subsets. We call differences between Radon measures charges, with upper, lower and total variations $\nu^+:=\sup\{\nu,0\}$, $\nu^-:=(-\nu)^+$ and $|\nu|:=\nu^++\nu^-$, respectively. As in the preface to [26], if the integral of a function with respect to a measure $\mu$ exists and takes a value in $\overline{\mathbb{R}}$, then we call this function $\mu$-integrable, and if, moreover, this integral is finite, that is, takes its value in $\mathbb{R}$, then the function is $\mu$-summable. We let $\mathfrak{m}_2$ denote the planar Lebesgue measure (or the two-dimensional Hausdorff measure) on $\mathbb{C}$ and $\mathfrak{m}_1$ denote the one-dimensional Hausdorff measure on $\mathbb{C}$, whose restriction to $\mathbb{R}$ is the linear Lebesgue measure on $\mathbb{R}$ and whose restriction to the image of a Lipschitz arc or curve is the arclength measure on this arc or curve (see [26], Ch. 5, § 2.1 and § 3.3.4A, and [9], § 5, Definition 3 and § 6.5.1). With a subharmonic function $u\colon D\to \overline{\mathbb{R}}$ such that $u\neq-\infty$ we associate the Radon measure — or Riesz mass distribution (see [27], Ch. 3, § 3.7, and [8], Ch. 3, § 3) where $\Delta$ is the Laplace operator on $D$ in the sense of distribution theory. By definition, the Riesz mass distribution of the subharmonic function $u=-\infty$ on $D$ is the exterior measure equal to $+\infty$ at each nonempty subset of $D$.Let $\varDelta$ be a Borel measure (or charge) on $\overline D(R)$. We let
$$
\begin{equation}
\varDelta^{\mathfrak{r}}\colon t\underset{t\in [0,R]}{\longmapsto} \varDelta\bigl(\overline D(t)\bigr)
\end{equation}
\tag{1.11}
$$
denote the counting radial function for $\varDelta$. If $s\colon \mathbb{R}\to \mathbb{R}$ is a bounded $2\pi$-periodic function with $|\varDelta|$-measurable composition
$$
\begin{equation}
s\circ \arg\colon z\underset{0\neq z\in \mathbb{C}}{\longmapsto} s(\arg z), \qquad (s\circ \arg)(0)\overset{(1.7)}{:=} \|s\|_{\mathbb{R}}\overset{(1.5)}{=}\sup_{\theta\in \mathbb{R}}\bigl|s(\theta)\bigr|,
\end{equation}
\tag{1.12}
$$
then by the counting radial-angular function with weight $s$ for $\varDelta$ [15]–[18] we mean the function $\varDelta^{\mathfrak{ra}(s)}$ on $[0,R]\subset \mathbb{R}$ defined by
$$
\begin{equation}
\varDelta^{\mathfrak{ra}(s)}(t)\underset{t\in [0,R]}{:=} \iint_{\overline D(t)} (s\circ \arg) \,\mathrm{d} \varDelta = \iint_{|z|\leqslant t} s(\arg z) \,\mathrm{d} \varDelta(z).
\end{equation}
\tag{1.13}
$$
In particular, for $s=1$ this is the counting radial function $\varDelta^{\mathfrak{r}}$ from (1.11). 1.2.5. Some named classes of generalized convex functionsAs usual, a function $F\colon I\to \mathbb{R}$ is convex on an interval $I\subset \mathbb{R}$ if for any pair of points $a,b\in I$ and pair of numbers $c_1,c_2\in \mathbb{R}$ the inequalities $F(x)\leqslant c_1x+c_2$ for $x:=a$ and $x:=b$ imply the analogous inequality for each $x\in [a,b]$. A function $F\colon I\to \mathbb{R}$ is convex with respect to the logarithm $\log$, or $\log$-convex for short, on an interval $I\subset \mathbb{R}^+$ if for any $a,b\in I$ and $c_1,c_2\in \mathbb{R}$ the inequalities $F(x)\leqslant c_1\log x+c_2$ for $x:=a$ and $x:=b$ imply the analogous inequality for each $x\in [a,b]$. For $p\in \mathbb{R}^+$ a function $s\colon \mathbb{R}\to \mathbb{R}$ is said to be $p$-trigonometrically convex on $\mathbb{R}$ [1], [2], [10]–[12] if for all pairs $a\leqslant b<a+\pi/p$ and $c_1,c_2\in \mathbb{R}$ it follows from the inequalities $s(x)\leqslant c_1\cos px+c_2\sin px$ for $x:=a$ and $x:=b$ that the analogous inequality holds for each $x\in [a,b]$. On the other hand $1$-trigonometrically convex functions are often called just trigonometrically convex. In what follows we need only $2\pi$-periodic $p$-trigonometrically convex functions. Each $0$-trigonometrically convex function that is $2\pi$-periodic on $\mathbb{R}$ is a constant. We denote by $p\text{-}\mathrm{trc}$ the class of all $2\pi$-periodic $p$-trigonometrically convex functions on $\mathbb{R}$, and
$$
\begin{equation}
p\text{-}\mathrm{trc}^+:=\bigl\{s\in p\text{-}\mathrm{trc}\mid s\geqslant 0\text{ on $\mathbb{R}$} \bigr\} \subset p\text{-}\mathrm{trc}
\end{equation}
\tag{1.14}
$$
denotes the subclass of all positive functions in $p\text{-}\mathrm{trc}$. For $0<p\in \mathbb{R}^+$ we say that a function $F\colon I\to \mathbb{R}$ on an interval $I\subset \mathbb{R}^+$ is $p$-power convex on $I$ [28]–[30] if for any pairs $a,b\in I$ and $c_1,c_2\in \mathbb{R}$ the inequalities $F(x)\leqslant c_1x^p+c_2x^{-p}$ for $x:=a$ and $x:=b$ imply the analogous inequality for each $x\in [a,b]$. Furthermore, we call $1$-power convex functions just power convex. By definition, a function $F$ on the interval $I\subset \mathbb{R}$ is called $0$-power convex if and only if it is $\log$-convex. In § 4 we discuss the properties of such generalized convex functions. In particular, such a function $F$ on an interval $I$ has a left-continuous left derivative and a right-continuous right derivative $F_{\mathrm{lh}}'$ and $F_{\mathrm{rh}}'$, respectively, everywhere on $I$, which coincide everywhere on $I$ outside some at most countable set. The function $F$ is differentiable outside this set, and the derivatives $F_{\mathrm{lh}}'$ and $F_{\mathrm{rh}}'$ are Riemann integrable on closed subintervals of $I$. 1.2.6. Subharmonic $\varrho$-homogeneous functions on the planeA function $M\colon \mathbb{C}\to \mathbb{R}$ is said to be homogeneous of degree $\varrho \in \mathbb{R}^+$, or $\varrho$-homogeneous for short, if $M(tz)=t^{\varrho}M(z)$ for each $t\in \mathbb{R}^+$ and all $z\in \mathbb{C}$. A subharmonic function $M$ is $\varrho$-homogeneous if and only if for some function $h\in \varrho\text{-}\mathrm{trc}$ we have the representation $M(z)\underset{z\in \mathbb{C}}{=}h(\arg z)|z|^{\varrho}$ (see [12], Ch. 1, Theorem 9.12), or in polar coordinates
$$
\begin{equation}
M(te^{i\theta})=h(\theta)t^{\varrho} \quad\text{for all } t\in \mathbb{R}^+ \text{ and }\theta\in \mathbb{R}, \quad\text{where } h\in \varrho\text{-}\mathrm{trc}.
\end{equation}
\tag{1.15}
$$
Unless otherwise stated, we treat the Stieltjes (Riemann–Stieltjes or Lebesgue–Stieltjes) integral over an interval in $\overline{\mathbb{R}}$ with endpoints $r<R$ with respect to a function $g$ of bounded variation on this interval, as the following integral over the interval $(r,R]\subset \overline{\mathbb{R}}$:
$$
\begin{equation}
\int_r^R \dots \,\mathrm{d} g:=\int_{(r,R]} \dots \,\mathrm{d} g.
\end{equation}
\tag{1.16}
$$
If $h\in \varrho\text{-}\mathrm{trc}$, then the generalized function — or, more precisely, Radon measure — $h''+\varrho^2h\geqslant 0$ is positive on $\mathbb{R}$, which is equivalent (see [10], Ch. 4, Theorem 24, or [11], Ch. 4, Theorem 4.1) to the increase of the function
$$
\begin{equation*}
h'_{\mathrm{rh}}(\theta)+\varrho^2\int_{\theta_0}^{\theta}h(\varphi)\,\mathrm{d} \varphi \quad\text{with respect to } \theta\in \mathbb{R} \quad\text{for fixed } \theta_0.
\end{equation*}
\notag
$$
Here the second term is absolutely continuous, and therefore the first term is a function of bounded variation. Hence the value of the measure $h''+\varrho^2h$ at an interval $(\theta_0,\theta]\subset \mathbb{R}$ with endpoints $\theta_0<\theta$ is expressed by the formula
$$
\begin{equation*}
(h''+\varrho^2h)\bigl((\theta_0,\theta]\bigr) =h'_{\mathrm{rh}}(\theta)- h'_{\mathrm{rh}}(\theta_0)+\varrho^2\int_{\theta_0}^{\theta}h(\varphi)\,\mathrm{d} \varphi.
\end{equation*}
\notag
$$
Moreover, the Riesz mass distribution of the subharmonic function (1.15) is factored into the $\otimes$-measure product of the radial and angular components. In terms of the densities of measures this can be expressed as
$$
\begin{equation}
(\,\mathrm{d}\varDelta_M)(te^{i\theta})=t^{\varrho-1}\,\mathrm{d} t\otimes \frac{1}{2\pi}\,\mathrm{d} \biggl(h'_{\mathrm{rh}}(\theta)+\varrho^2\int^{\theta}h(\varphi)\,\mathrm{d} \varphi\biggr).
\end{equation}
\tag{1.17}
$$
For an arbitrary bounded $2\pi$-periodic $(h''+\varrho^2h)$-measurable function $s\colon \mathbb{R}\to \mathbb{R}$ the radially angular density (1.12), (1.13) with weight $s$ of the Riesz mass distribution of the function (1.15) is obviously equal to zero for $\varrho=0$, while for $\varrho >0$ it can be calculated by the formula
$$
\begin{equation}
\varDelta_M^{\mathfrak{ra}(s)}(r)\overset{(1.17)}{\underset{r\geqslant 0}{=}} \frac{1}{\varrho} \, r^{\varrho} \, \frac{1}{2\pi}\int_0^{2\pi}s(\theta)\bigl(\mathrm{d} h'_{\mathrm{rh}}(\theta)+\varrho^2h(\theta)\,\mathrm{d} \theta\bigr).
\end{equation}
\tag{1.18}
$$
We denote the arising Stieltjes integral times $1/2$ by
$$
\begin{equation}
\mathrm{A}_{\varrho}(s,h):=\frac{1}{2}\int_0^{2\pi} s(\theta)\bigl(\mathrm{d} h'_{\mathrm{rh}}(\theta)+\varrho^2h(\theta)\,\mathrm{d} \theta\bigr).
\end{equation}
\tag{1.19}
$$
We call the quantity $\mathrm{A}_{\varrho}(s,h)$ the $\varrho$-mixture of $s$ with the $\varrho$-trigonometrically convex function $h$. A motivation for this name is the close connection between the integral (1.19) and Minkowski’s mixed area (see [31], § 8, [32], Ch. 1, [15], § 3, Remark 2, and § 4, [17], Ch. 3, § 3.3.1, [28], and [29]). 1.3. The preceding resultsWe present the detailed statements of only those results on the main problem that, apart from the distribution of points in radius, also take account of their distribution in argument. For unification we state them using where possible the terminology and notation from § 1.2, so the formulations can slightly differ from the original ones. The most significant previous results on the main problem for functions on $\mathbb{C}$ were established for subharmonic $\varrho$-homogeneous majorants (1.15). The following uniqueness result (Theorem 1.1) was obtained by Grishin and Sodin, in a joint paper of 1988, for entire functions $f$ of finite (upper) type
$$
\begin{equation}
\limsup_{z\to \infty}\dfrac{\log |f(z)|}{|z|^\varrho}<+\infty
\end{equation}
\tag{1.20}
$$
with respect to order $\varrho\in \mathbb{R}^+$ such that the $2\pi$-periodic growth indicator
$$
\begin{equation}
h_f\colon \theta\underset{\theta\in \mathbb{R}}{\longmapsto} \limsup\frac{\log|f(re^{i\theta})|}{r^{\varrho}}\in \mathbb{R}
\end{equation}
\tag{1.21}
$$
with respect to order $\varrho$ is a $\varrho$-trigonometrically convex function. Theorem 1.1 (see [13], Theorem 6.2). Let $f$ be an entire function vanishing on $Z$ that has a finite upper type with growth indicator $h_f\leqslant h\in \varrho\text{-}\mathrm{trc}$. If for $s\in \varrho\text{-}\mathrm{trc}^+$ and the $\varrho$-mixture (1.19) the strict inequality holds, then $f=0$ is the zero function. Theorem 1.1 was significantly developed in our papers (see [14], § 7, and [15], § 3, Theorem 3.2), in which as the test function $s$ we took arbitrary $p$-trigonometrically convex functions $s\geqslant 0$ with an arbitrary exponent $p\in \mathbb{R}^+$, not related to the order $\varrho$ of $f$. In this setting Theorem 3.2 in [15], § 3, was obtained for arbitrary $p\leqslant \varrho$ and straight away in the multidimensional subharmonic version, for functions on the $n$-dimensional real space $\mathbb{R}^n$. Some results in our papers were stated as equivalent dual theorems on the completeness of exponential or more general parametrized systems of entire functions in spaces of holomorphic functions on domains or compact subsets of $\mathbb{C}$ (see [15], Theorem 4.1, [16], Theorems A and 2.1, [17], Theorems 3.3.5 and 3.3.9) or of $\mathbb{C}^n$ (see [16], Theorem 3.1, and [17], Theorem 4.2.7). These uniqueness theorems, together with adaptations of completeness theorems, can be formulated as follows in the form of a final summary for $\mathbb{C}$. Theorem 1.2 (see [14]–[17]). Let $f$ be an entire function as in Theorem 1.1, let $p\in \mathbb{R}^+$, and let $\mathrm{A}_{\varrho}(s,h)$ be the $\varrho$-mixture from (1.19) of some $p$-trigonometrically convex positive function $s \in p\text{-}\mathrm{trc}^+$ with a function $h\in \varrho\text{-}\mathrm{trc}$. If at least one of the following three conditions is satisfied:
Theorems 1.1 and 1.2 are only interesting in the case of finite upper density
$$
\begin{equation}
\limsup_{t\to+\infty}\frac{Z^{\mathfrak{r}}(t)}{t^{\varrho}} \overset{(1.9)}{<}+\infty
\end{equation}
\tag{1.26}
$$
of the distribution of points $Z$ with respect to order $\varrho$, because otherwise any entire function of order $\varrho$ and finite type that vanishes on $Z$ is zero. In the case (1.26), on the basis of [33], § 1, [34], Theorem 1, [35], Proposition 6, and [3], § 3.2, Proposition 1, we can show that the double limit on the left-hand side of (1.24) is never less that the limit on the left-hand side of (1.22), and there are cases when they are distinct ([17], Example 3.2.1). Thus, already in the framework of inequality (1.24) in condition (ii), Theorem 1.2 is stronger than Theorem 1.1. Below we generalize Theorem 1.2 significantly by using $p$-power convex functions in place of particular power functions in Theorem 2.3 on uniqueness and in its corollaries. Before that, in Theorem 2.2 we find analogues of condition (1) that are more comprehensible in their meaning as well as in their form. Finally, we present one of the criteria mentioned in § 1.1. In [24] we denoted by $\mathrm{Pot}_0^{+1}$ the class of positive subharmonic functions $q$ on $\mathbb{C}$ such that $q(0)=0$, with the seminormalization
$$
\begin{equation}
\limsup_{z\to \infty}\frac{q(z)}{\log|z|}\leqslant 1.
\end{equation}
\tag{1.27}
$$
Theorem 1.3 ([24], Theorem 3). Let $M=M^{\mathrm{up}}-M_{\rm{low}}$ be the difference of two subharmonic functions $M^{\mathrm{up}}$ and $M_{\rm{low}}$ such that $M^{\mathrm{up}}(0)+M_{\mathrm{low}}(0)\neq -\infty$, with the Riesz charge distribution $\varDelta_M=\varDelta_{M^{\mathrm{up}}}-\varDelta_{M_{\mathrm{low}}}$, which satisfies condition (1.4) for some $P\geqslant 0$. A distribution $Z$ of points on $\mathbb{C}$ is a uniqueness distribution with respect to $M$ if and only if The sufficiency part of this uniqueness criterion will be developed in Theorem 2.3 on uniqueness, by involving functions with separated dependence on the radius and argument and a more general normalization than the logarithmic (semi)normalizations by one covered by (1.27). § 2. Distribution of zeros of entire function with subharmonic majorant2.1. Arbitrary subharmonic majorants $M$The following result is a clue to the distribution of zeros of entire functions. Theorem 2.1. Let $Z$, $Z(0)=0$, be a distribution of points on $\mathbb{C}$, and let $M$ be a subharmonic function on $\mathbb{C}$ for which there exists an entire function $f$ vanishing on $Z$ such that $f(0)\neq 0$ and $\log |f|\leqslant M$ on $\mathbb{C}$. Fix the following objects arbitrarily: Then the following inequality holds:
$$
\begin{equation}
\int_r^R \bigl(Z^{\mathfrak{ra}(s)}(t)-\varDelta_M^{\mathfrak{ra}(s)}(t)\bigr) (-F'_{\mathrm{rh}}(t))\,\mathrm{d} t \leqslant \|s\|_{\mathbb{R}} Q_{p,F}(r)\bigl(M^{\circ r}-\log|f(0)|\bigr),
\end{equation}
\tag{2.2}
$$
where $(-F'_{\mathrm{rh}}(t))\underset{t\in [r,R)}{\geqslant} 0$, $\|s\|_{\mathbb{R}}$ and $M^{\circ r}-\log|f(0)|\overset{(1.1)}{\geqslant} 0$ are positive and
$$
\begin{equation}
Q_{p,F}(r):=p\bigl(F(r)-F(R)\bigr)-rF'_{\mathrm{rh}}(r)\in \mathbb{R}^+.
\end{equation}
\tag{2.3}
$$
If $F$ in (II) is a positive decreasing $ p$-power convex function $F$ on $(0, R)$ without condition (2.1), but such that the limit
$$
\begin{equation}
{\lim}_0^pF:=\limsup_{0<t\to 0}t^pF(t) \quad\textit{for } p>0 \quad\textit{or}\quad {\lim}_0^0F:=\limsup_{0<t\to 0}\frac{F(t)}{\log (1/t)} \quad\textit{for } p=0
\end{equation}
\tag{2.4}
$$
is finite, then for all $r\in (0,R)$ where $\check{p}$ is defined in terms of $p$ by We deduce Theorem 2.1 at the end of § 5.2 from its subharmonic version, Theorem 5.2, which we also prove there. We state and prove our main theorem ( Theorem 5.1), from which Theorem 5.2 is derived, in § 5.1. In the general, merely ‘radial’ case, for $p=0$ and $\check p\overset{(2.6)}{=}1$ we immediately obtain the following result. Corollary 2.1. Taking $p:=0$ and $s:=1$ under the assumptions of Theorem 2.1 yields In particular, if $F$ is a positive decreasing $\log$-convex function on $(0,R)$ and ${\lim}_0^0F\overset{(2.4)}{<}+\infty$, then for all $r\in (0,R)$ we have The last, merely radial case (2.7) was considered before only for a radial majorant $M$ and a continuously differentiable $\log$-convex function $F$, in the form of a uniqueness theorem, in our joint paper with Tamindarova (see [36], Theorem 1), containing also relevant radial results for subharmonic and entire functions of several variables. From the second part of Theorem 2.1 with final inequality (2.5) for a function $M$ of the form (1.15) we can deduce as special cases all uniqueness theorems from [13] (the uniqueness theorem), [14] (the uniqueness theorem) and [15] (Theorem 4.1). In their statements and, accordingly, in Theorems 1.1 and 1.2 a single class of positive decreasing $p$-power convex functions is used, namely,
$$
\begin{equation}
F\colon t\underset{t\in (0,R)}{\longmapsto} \biggl(\frac{R}{t}\biggr)^p-\biggl(\frac{t}{R}\biggr)^p =\frac{1}{R^p}\frac{R^{2p}-t^{2p}}{t^p},
\end{equation}
\tag{2.8}
$$
with the opposite of the derivative equal to
$$
\begin{equation}
-F'(t)=\frac{p}{t}\biggl(\biggl(\frac{R}{t}\biggr)^p+\biggl(\frac{t}{R}\biggr)^p\biggr).
\end{equation}
\tag{2.9}
$$
In the integral (1.23) in condition (i) this derivative is explicit in the integrand; in the integral (1.24) in condition (ii) the first term in parentheses in (2.9) is present in the integrand with a coefficient, while the second is dropped as negligible; and in the second integral in (1.25), in condition (ii) the same term from (2.9) participates for the limit value of $R:=+\infty$. The characteristic property of the function (2.8) as a $p$-power convex function was not revealed in [13]–[15]. On the other hand, the results there were only asymptotic as $R\to +\infty$, and the case when $r$ can take values between $0$ and $R$, is not covered by conditions (i) and (iii) in Theorem 1.2. We do not present here the derivation of Theorem 1.2 from Theorem 2.1. Instead, in § 2.2 we present Theorem 2.2, which is a development of Theorems 1.1 and 1.2 for $p\leqslant \varrho$. In place of the functions (2.8), which satisfy (2.9), we take some simpler functions. Example 2.1. The function $ F\colon t\underset{t\in (0,R)}{\longmapsto}\log^+ (R/t)$, where ${\lim}_0^0F=1$, satisfies the assumptions of Corollary 2.1, and the conclusion (2.7) of the corollary coincides with (1.3). Example 2.2. For each choice of $p>0$ the decreasing positive function $F\colon t\underset{t>0}{\longmapsto} 1/ t^p$ with limit value ${\lim}_0^pF\overset{(2.4)}{=}\lim_{0<t\to 0} t^p(1/t)^p=1$ at zero and $F'(t)=-pt^{-p-1}$ is an example of a $p$-power convex function on $(0,+\infty)$: this can easily be seen from its definition or from Proposition 4.2. Applying Theorem 2.1, namely, (2.4)–(2.6), to the functions $F$ from Examples 2.1 and 2.2 we immediately obtain the following result. Corollary 2.2. Under the assumption of Theorem 2.1, when only the objects from (I) are fixed, for all $0<r<R<+\infty$ 2.2. Subharmonic $\varrho$-homogeneous majorants $M$In the case of subharmonic $\varrho$-homogeneous majorants $M$, which we discussed in § 1.2.6 and which have the form (1.15), using (1.18) and (1.19) we can write
$$
\begin{equation}
\varDelta_M^{\mathfrak{ra}(s)}(t)\overset{(1.19)}{\underset{t\geqslant 0}{=}} \frac{1}{\pi \varrho} \mathrm{A}_{\varrho}(s,h)t^{\varrho}.
\end{equation}
\tag{2.11}
$$
Hence the subtrahend on the left-hand side of (2.10) has the expression
$$
\begin{equation}
\int_r^R\frac{\varDelta_M^{\mathfrak{ra}(s)}(t)}{t^{p+1}}\,\mathrm{d} t \overset{(2.11)}{=} \frac{\mathrm{A}_{\varrho}(s,h)}{\pi\varrho}\int_r^R \frac{1}{t^{p+1}}t^{\varrho}\,\mathrm{d} t,
\end{equation}
\tag{2.12}
$$
where the right-hand side is equal to
$$
\begin{equation}
\frac{\mathrm{A}_{\varrho}(s,h)}{\pi\varrho}\cdot \begin{cases} \log\dfrac{R}{r}&\text{for } p=\varrho, \\ \dfrac{R^{\varrho-p}-r^{\varrho-p}}{\varrho-p} &\text{for } 0\leqslant p\neq\varrho. \end{cases}
\end{equation}
\tag{2.13}
$$
Corollary 2.3. Let $f\neq 0$ be an entire function vanishing on a distribution of points $Z$ such that $Z(0)=0$, and assume that for some $h\in \varrho\text{-}\mathrm{trc}$ and $c\in \mathbb{R}$ Then there exists $C\in \mathbb{R}^+$ such that for $p\in \mathbb{R}^+$ and any function $s\in p\text{-}\mathrm{trc}^+$, for all $1\leqslant r<R<+\infty$
$$
\begin{equation}
r^{p-\varrho}\int_r^R\frac{Z^{\mathfrak{ra}(s)}(t)}{t^{p+1}} \,\mathrm{d} t\leqslant \frac{\mathrm{A}_{\varrho}(s,h)}{\pi\varrho}\cdot \begin{cases} \log\dfrac{R}{r}&\textit{for } p=\varrho, \\ \dfrac{(R/r)^{\varrho-p}-1}{\varrho-p} &\textit{for } p\neq\varrho , \end{cases} +C\|s\|_{\mathbb{R}}.
\end{equation}
\tag{2.15}
$$
Proof. In (2.14) we can assume that $c=0$ because we can go over to the entire function $e^{-c}f\neq 0$ with the same properties and with a majorizing function ${M\geqslant \log|f|}$ of the form (1.15), such that
$$
\begin{equation}
M^{\circ r}\overset{(1.1)}{=}\frac{1}{2\pi}\int_0^{2\pi}h(\theta) r^{\varrho}\,\mathrm{d} \theta\leqslant \|h\|_{\mathbb{R}} r^{\varrho}.
\end{equation}
\tag{2.16}
$$
Moreover, if $n=\operatorname{Zero}_f(0)>0$, then the ratio of $f$ and $Nz^n$, where $N>0$ is sufficiently large, has the same properties. Hence we can assume that $f(0)\neq 0$ in what follows. Then the assumptions of Theorem 2.1, and therefore those of Corollary 2.2 are fulfilled. The assertion of Corollary 2.2 means in view of (2.12) and (2.13) that for all $0<r<R<+\infty$ we have the inequality
$$
\begin{equation*}
\int_r^R\frac{Z^{\mathfrak{ra}(s)}(t)}{t^{p+1}}\,\mathrm{d} t\leqslant \frac{\mathrm{A}_{\varrho}(s,h)}{\pi\varrho}\cdot \begin{cases} \log\dfrac{R}{r}&\text{for } p=\varrho, \\ \dfrac{R^{\varrho-p}-r^{\varrho-p}}{\varrho-p} &\text{for } p\neq\varrho, \end{cases} +2\|s\|_{\mathbb{R}}\frac{M^{\circ r}-\log|f(0)|}{r^p},
\end{equation*}
\notag
$$
where for the last fraction on the right-hand side, for all $r\geqslant 1$, we have
$$
\begin{equation*}
\frac{M^{\circ r}-\log|f(0)|}{r^p}\underset{r\geqslant 0}{\overset{(2.16)}{\leqslant}} \frac{\|h\|_{\mathbb{R}} r^{\varrho}-\log|f(0)|}{r^p}\underset{r\geqslant 1}{\leqslant} \frac{\|h\|_{\mathbb{R}} r^{\varrho}+r^{\varrho}\log^-|f(0)|}{r^p},
\end{equation*}
\notag
$$
so that the last term on the right-hand side satisfies
$$
\begin{equation*}
2\|s\|_{\mathbb{R}}\frac{M^{\circ r}-\log|f(0)|}{r^p}\leqslant 2\bigl(\|h\|_{\mathbb{R}} +\log^-|f(0)|\bigr)\|s\|_{\mathbb{R}}r^{\varrho-p} \quad\text{for all } r\geqslant 1.
\end{equation*}
\notag
$$
Thus, for $C:=2\bigl(\|h\|_{\mathbb{R}} +\log^-|f(0)|\bigr)$ we obtain
$$
\begin{equation*}
\int_r^R\frac{Z^{\mathfrak{ra}(s)}(t)}{t^{p+1}}\leqslant \frac{\mathrm{A}_{\varrho}(s,h)}{\pi\varrho}\cdot \begin{cases} \log\dfrac{R}{r}&\text{for } p=\varrho, \\ \dfrac{R^{\varrho-p}-r^{\varrho-p}}{\varrho-p} &\text{for } 0\leqslant p\neq\varrho, \end{cases} +C\|s\|_{\mathbb{R}}r^{\varrho-p}
\end{equation*}
\notag
$$
for all $1\leqslant r<R<+\infty$. Dividing both sides by $r^{\varrho-p}$ we arrive at the required inequality (2.15), which completes the proof. Remark 2.1. Corollary 2.3 is convenient for the illustration of the sharpness of our results here. For example, the left-hand side of (2.15) is ‘sensitive’ for $p\leqslant \varrho$ to changes even by one in the multiplicities $Z(z)$ for $z\neq 0$, because for ${s_z\!:=\!s(\arg z)\!>\!0}$ we clearly have Various other examples for $\varrho$-homogeneous subharmonic functions (1.15) can be constructed on the basis of subtle results on approximations of subharmonic functions by the logarithm of the modulus of an entire function, such as, for instance, in [37] by Lutsenko and Yulmukhametov. 2.3. Uniqueness theoremsThe following uniqueness theorem is a development of Theorems 1.1 and 1.2. Theorem 2.2 (on uniqueness). Let $f$ be an entire function vanishing on $Z$ that has a finite upper type with growth indicator $h_f\leqslant h\in \varrho\text{-}\mathrm{trc}$, as in Theorems 1.1 and 1.2. Assume that at least one of the following two conditions is satisfied:
Proof. By assumption, for each $\varepsilon >0$ there exists $c_{\varepsilon}\geqslant 0$ such that in the polar coordinates $t:=|z|, \theta\in \arg z$ we have
$$
\begin{equation}
\log |f(te^{\theta})|\leqslant \bigl(h(\theta)+\varepsilon\bigr)t^{\varrho}+c_{\varepsilon} \quad\text{for all } z=te^{i\theta}\in \mathbb{C}.
\end{equation}
\tag{2.21}
$$
In addition, it is easy to see from the definition (1.19) of the $\varrho$-mixture of a function $s\in p\text{-}\mathrm{trc}^+$ with a $\varrho$-trigonometrically convex function $h+\varepsilon$ that
$$
\begin{equation}
\lim_{0<\varepsilon \to 0} \mathrm{A}_{\varrho}(s,h+\varepsilon)=\mathrm{A}_{\varrho}(s,h).
\end{equation}
\tag{2.22}
$$
In the proof we assume that $f\neq 0$, and then using Corollary 2.3 we establish inequalities contradicting the corresponding inequalities in (i) and (ii).
From (2.21) and (2.15) for $r:=1$, for the entire function $f\neq 0$ we obtain
$$
\begin{equation}
\int_1^R\frac{Z^{\mathfrak{ra}(s)}(t)}{t^{p+1}} \,\mathrm{d} t\leqslant \frac{\mathrm{A}_{\varrho}(s,h+\varepsilon)}{\pi\varrho}\cdot \begin{cases} \log R &\text{for } p=\varrho, \\ \dfrac{R^{\varrho-p}-1}{\varrho-p} &\text{for } p\neq\varrho, \end{cases} +C\|s\|_{\mathbb{R}},
\end{equation}
\tag{2.23}
$$
where $C$ is independent of $p$, $s$ and $R>1$, but depends on $\varepsilon >0$, and for an arbitrary $a>1$, for $R:=ar$ we obtain
$$
\begin{equation}
r^{p-\varrho}\int_r^{ar}\frac{Z^{\mathfrak{ra}(s)}(t)}{t^{p+1}} \,\mathrm{d} t\leqslant \frac{\mathrm{A}_{\varrho}(s,h+\varepsilon)}{\pi\varrho}\cdot \begin{cases} \log a&\text{for } p=\varrho, \\ \dfrac{a^{\varrho-p}-1}{\varrho-p} &\text{for } p\neq\varrho, \end{cases} +C\|s\|_{\mathbb{R}},
\end{equation}
\tag{2.24}
$$
where $C$ is independent of $p$, $s$, $a>1$ and $r\geqslant 1$, although it depends on $\varepsilon >0$. The right-hand side of (2.24) is independent of $r$, so taking the upper limit as $r\to +\infty$ we obtain
$$
\begin{equation}
\begin{aligned} \, &\limsup_{r\to +\infty} \frac{1}{r^{\varrho-p}} \int_r^{ar}\frac{Z^{\mathfrak{ra} (s)}(t)}{t^{p+1}}\,\mathrm{d} t \nonumber \\ &\qquad\leqslant \frac{\mathrm{A}_{\varrho}(s,h+\varepsilon)}{\pi\varrho}\cdot \begin{cases} \log a&\text{for } p=\varrho, \\ \dfrac{a^{\varrho-p}-1}{\varrho-p} &\text{for } p\neq\varrho, \end{cases} +C\|s\|_{\mathbb{R}} \end{aligned}
\end{equation}
\tag{2.25}
$$
where $C\in \mathbb{R}^+$ has the same properties.
Now consider (i) and (ii) for $0\leqslant p\leqslant \varrho$ and $r:=1$ and assume that (2.19) or (1.22) holds. Dividing both sides of (2.23) by $\log R$ for $p=\varrho$ and by $R^{\varrho-p}$ for ${0\leqslant p<\varrho}$ and taking the upper limit as $R\to +\infty$ we obtain
$$
\begin{equation}
\limsup_{R\to +\infty}\frac{1}{\log R}\int_1^R\frac{Z^{\mathfrak{ra}(s)}}{t^{\varrho+1}}\,\mathrm{d} t \leqslant \frac{\mathrm{A}_{\varrho}(s,h+\varepsilon)}{\pi \varrho} \quad\text{for } p=\varrho
\end{equation}
\tag{2.26}
$$
and
$$
\begin{equation}
\limsup_{R\to +\infty} \frac{1}{R^{\varrho-p}} \int_1^R\frac{Z^{\mathfrak{ra} (s)}(t)}{t^{p+1}}\,\mathrm{d} t \leqslant \frac{\mathrm{A}_{\varrho}(s,h+\varepsilon)}{\pi \varrho(\varrho-p)} \quad\text{for } 0\leqslant p<\varrho,
\end{equation}
\tag{2.27}
$$
respectively. Taking the limits as $0<\varepsilon \to 0$ in (2.26) and (2.27), in view of (2.22) we obtain inequalities contradicting (1.22) and (2.19). This proves Theorem 2.2 under assumptions (1.22) and (2.19).
Consider (i) and (ii) for $0\leqslant p\leqslant \varrho$ and $R:=ar$, and assume that (1.24) or (2.20) holds. The quantities $\log a$ and $a^{\varrho-p}$ for $p<\varrho$ tend to $+\infty$ as $1<a\to +\infty$. We divide both sides of (2.25) by $\log a$ for $p=\varrho$ or by $a^{\varrho-p}$ for $p<\varrho$. Now taking the upper limit as $1<a\to +\infty$ yields the inequalities
$$
\begin{equation*}
\limsup_{1<a\to +\infty}\frac{1}{\log a}\limsup_{r\to +\infty} \int_r^{ar}\frac{Z^{\mathfrak{ra} (s)}(t)}{t^{\varrho+1}}\,\mathrm{d} t \leqslant \frac{\mathrm{A}_{\varrho}(s,h+\varepsilon)}{\pi \varrho}\quad\text{for } p=\varrho
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\limsup_{1<a\to +\infty}\frac{1}{a^{\varrho-p}-1} \limsup_{r\to +\infty} \frac{1}{r^{\varrho-p}} \int_r^{ar}\frac{Z^{\mathfrak{ra} (s)}(t)}{t^{p+1}}\,\mathrm{d} t \leqslant \frac{\mathrm{A}_{\varrho}(s,h+\varepsilon)}{\pi \varrho(\varrho-p)} \quad\text{for } p<\varrho.
\end{equation*}
\notag
$$
Their left-hand sides are independent of $\varepsilon >0$. Hence letting $\varepsilon>0$ tend to zero we obtain inequalities contradicting (1.24) and (2.20), respectively. This proves the corollary under assumptions (1.24) and (2.20).
Theorem 2.2 is proved. For $p\in \mathbb{R}^+$ we let $p\text{-}\mathrm{pwc}(r,R)$ denote the class of $ p$-power convex functions on the interval $(r,R)\subset \mathbb{R}^+$, as defined in § 1.2.5; let
$$
\begin{equation*}
p\text{-}\mathrm{pwc}^+(r,R):=\bigl\{F\in p\text{-}\mathrm{pwc}(r,R)\mid F\geqslant 0 \text{ on $(r,R)$}\bigr\}\subset p\text{-}\mathrm{pwc}(r,R)
\end{equation*}
\notag
$$
be the subclass of positive functions from $p\text{-}\mathrm{pwc}(r,R)$, $p\text{-}\mathrm{pwc}^{\downarrow}(r,R)$ be the subclass of decreasing functions on $(r,R)$, and set $F(R):=\lim_{R>t\to R}F(t)$; let
$$
\begin{equation}
p\text{-}\mathrm{pwc}^{+\downarrow}(r,R)=p\text{-}\mathrm{pwc}^+(r,R)\cap p\text{-}\mathrm{pwc}^{\downarrow}(r,R)
\end{equation}
\tag{2.28}
$$
be the subclass of positive decreasing functions from $p\text{-}\mathrm{pwc}(r,R)$; let
$$
\begin{equation}
p\text{-}\mathrm{pwc}^{+\downarrow}_1(0,R):=\bigl\{F\in p\text{-}\mathrm{pwc}^{+\downarrow}(0,R)\mid {\lim}_0^pF\overset{(2.4)}{=}1\bigr\}
\end{equation}
\tag{2.29}
$$
be the subclass of functions in $p\text{-}\mathrm{pwc}^{+\downarrow}(0,R)$ normalized by one, in the sense of the limit value ${\lim}_0^pF$ at zero as defined in (2.4). Given a mass distribution $\varDelta$ on $\mathbb{C}$, apart from its radial counting function $r \overset{(1.11)}{\underset{r\in \mathbb{R}^+}{\longmapsto}} \varDelta^{\mathfrak{r}}(r)$, we introduce its averaged radial counting function
$$
\begin{equation}
\varDelta^{\mathfrak{r}\circ}\colon r\overset{(1.11)}{\underset{r\in \mathbb{R}^+}{\longmapsto}} \int_0^r\frac{\varDelta^{\mathfrak{r}}(t)}{t}\,\mathrm{d} t.
\end{equation}
\tag{2.30}
$$
The following results is the most general uniqueness theorem in this paper. Theorem 2.3 (uniqueness theorem). Let $Z$ be a point distribution such that ${Z(0)=0}$, let $\varDelta$ be a mass distribution on $\mathbb{C}$, and let the function $m\colon \mathbb{R}^+\to \mathbb{R}^+$ be bounded away from zero. If for some subclass $T_p\overset{(1.14)}{\subset} p\text{-}\mathrm{trc}^+$ the equality
$$
\begin{equation}
\begin{aligned} \, \notag &\sup_{0<r<R\in \mathbb{R}^+}\frac{1}{m(r)}\sup\biggl\{r^p\int_r^R \bigl(Z^{\mathfrak{ra}(s)}(t)-\varDelta^{\mathfrak{ra}(s)}(t)\bigr) \frac{-F'_{\mathrm{rh}}(t)}{\check{p}\|s\|_{\mathbb{R}}}\,\mathrm{d} t \biggm| \\ &\qquad p\in \mathbb{R}^+, \ (s, F)\in T_p\times p\textit{-}\mathrm{pwc}^{+\downarrow}_1(0,R), \ s\neq 0\biggr\} =+\infty \end{aligned}
\end{equation}
\tag{2.31}
$$
holds, where $T_p\times p\text{-}\mathrm{pwc}^{+\downarrow}_1(0,R)$ is the Cartesian product of the classes $T_p$ and $p\text{-}\mathrm{pwc}^{+\downarrow}_1(0,R)$ from (1.14) and (2.29), then $Z$ is a uniqueness distribution with respect to any subharmonic function $M$ on $\mathbb{C}$ that is bounded below in a neighbourhood of zero and satisfies the inequalities $\varDelta_M^{\mathfrak{ra}(s)}\overset{(1.13)}{\leqslant} \varDelta^{\mathfrak{ra}(s)}$ on $\mathbb{R}^+$ for all $s\in T_p$, as well as the constraint The equivalence in condition (2.32) is here an obvious — because ${M(0)>-\infty}$ — consequence of the Poisson–Jensen–Nevanlinna–Privalov formula (1.2). Theorem 2.3 is proved in § 5.3. Directly from Theorem 2.3 on uniqueness we obtain its ‘individualized’ version, involving a single pair of functions $s$ and $F$. In its statement the supremum in (2.31) and the normalization ${\lim}_0^pF=1$ are superfluous. Corollary 2.4. In the assumptions of Theorem 2.3 let $s\overset{(1.14)}{\in} p\text{-}\mathrm{trc}^+$ and $F\overset{(2.28)}{\in} p\text{-}\mathrm{pwc}^{+\downarrow}(0,+\infty)$ be a pair of function with finite limit ${\lim}_0^pF\overset{(2.4)}{<}+\infty$, and let Then $Z$ is a uniqueness distribution with respect to each subharmonic function $M$ on $\mathbb{C}$ as in Theorem 2.3 that satisfies $\varDelta_M^{\mathfrak{ra}(s)}\leqslant \varDelta^{\mathfrak{ra}(s)}$ on $\mathbb{R}^+$ with respect to the fixed function $s\in p\text{-}\mathrm{trc}^+$, provided that constraint (2.32) holds. The function $B\colon \mathbb{R}^+\to \overline{\mathbb{R}}$ has a finite type with respect to order $\varrho\in \mathbb{R}^+$ if
$$
\begin{equation*}
\sup_{R\in \mathbb{R}^+}\frac{B^+(R)}{1+R^{\varrho}}<+\infty.
\end{equation*}
\notag
$$
The function $M\colon \mathbb{C}\to \overline{\mathbb{R}}$ has a finite type with respect to order $\varrho\in \mathbb{R}^+$ if this holds for the function equal to its supremum over the circles
$$
\begin{equation*}
r\underset{r\in \mathbb{R}^+}{\longmapsto} \sup_{|z|=r}M(z)\underset{r\in \mathbb{R}^+}{\overset{(1.1)}{\geqslant}} M^{\circ r}.
\end{equation*}
\notag
$$
For an entire function $f$ and $M:=\log|f|$ we obtain just (1.20). The following simple facts are known about functions of finite type with respect to an order $\varrho\in \mathbb{R}^+$. Proposition 2.1. Let $\varrho\in \mathbb{R}^+$, let $\varDelta$ be a mass distribution on $\mathbb{C}$ and $M$ be a subharmonic function with Riesz mass distribution $\varDelta_M=\varDelta$. If at least one of the following three functions is a function of order $\varrho$ and finite type with respect to the variable $R$, for a fixed variable $r$ in (ii), (iii): then so are also the other two functions in (i)–(iii). A fortiori, if $M$ has order $\varrho$ and finite type, then so do all functions in (i)–(iii), although the converse is not true in general.If a radial counting function is a point or mass distribution of finite type with respect to order $\varrho$, then we can also say that this distribution has a finite upper density with respect to order $\varrho$, as in (1.26). Using Proposition 2.1 we can easily formulate the result of Theorem 2.3 in terms of the standard radial counting function $\varDelta^{\mathfrak{r}}$. Corollary 2.5. Under the assumptions of Theorem 2.3 or Corollary 2.4, if equality (2.31) or (2.33), respectively, holds for $m(r)\underset{r\in \mathbb{R}^+}{:=}1+r^\varrho$, then the assertion of Theorem 2.3 or Corollary 2.4, respectively, is valid after the replacement of condition (2.32) by the condition of finite upper density of the Riesz mass distribution $\varDelta_M$ with respect to order $\varrho$. Remark 2.2. Relaxing Theorem 2.3 on uniqueness and Corollaries 2.4 and 2.5 we can replace the outer suprema over $0<r<R\in \mathbb{R}^+$ in (2.31) and (2.33) by the upper limit as the pair $r<R$ tends to $+\infty$, or just as $R\to +\infty$, but then for fixed $r>0$. Precisely such weaker statements involving upper limits were used in Theorems 1.1 and 1.2, established previously and presented in § 1.3, as well as in Theorem 2.2. In particular, using such upper limits is expedient in the case when, in place of rigid constraints of the form $\log|f|\leqslant M$ in the main problem, for a strictly positive subharmonic majorizing function $M$ and a quantity $b\in \overline{\mathbb{R}}^+$ we consider much milder constraints of the form From Theorem 2.3 and Corollary 2.5, taking the class $T_p:= p\text{-}\mathrm{trc}^+$ and the mass distribution $\varDelta$ equal to $\varDelta_M$ in (1.17), from equality (1.18) and (1.19) we immediately obtain the following result. Corollary 2.6. Let $Z$ be a distribution of points on $\mathbb{C}$ such that $Z(0)=0$ and $h$ be a $2\pi$-periodic $\varrho$-trigonometrically convex function on $\mathbb{R}$. If
$$
\begin{equation*}
\begin{aligned} \, &\sup_{0< r<R<+\infty}\frac{1}{1+r^{\varrho}}\sup\biggl\{ r^p\int_r^R \biggl(Z^{\mathfrak{ra}(s)}(t)-\frac{\mathrm{A}_{\varrho}(s,h)}{\pi\varrho}t^{\varrho} \biggr) \frac{-F'_{\mathrm{rh}}(t)}{\check{p}\|s\|_{\mathbb{R}}}\,\mathrm{d} t \biggm| \\ &\qquad p\in \mathbb{R}^+, \ (s,F)\overset{(1.14),(2.29)}{\in} p\textit{-}\mathrm{trc}^+\times p\textit{-}\mathrm{pwc}^{+\downarrow}_1(0,R), \ s\neq 0\biggr\} =+\infty, \end{aligned}
\end{equation*}
\notag
$$
then $Z$ is a uniqueness distribution with respect to the function (1.15) on $\mathbb{C}$. We also present an individualized version of this corollary for one pair $(s,F)$. Corollary 2.7. Under the assumptions of Corollary 2.6 fix a pair of functions $s\in p\text{-}\mathrm{trc}^+$ and $F\overset{(2.29)}{\in} p\text{-}\mathrm{pwc}^{+\downarrow}(0,+\infty)$ с ${\lim}_0^pF<+\infty$ such that Then $Z$ is a uniqueness distribution with respect to the function (1.15) on $\mathbb{C}$. § 3. General inequalities for pairs of domains adjacent across an arc or a closed curveFor a subset $S$ of $ \mathbb{C}$ and $k\in \overline{\mathbb{N}}$ let $C^k(S)$ denote the class of real functions defined in a neighbourhood of $S$, with continuous partial derivatives of order $k\in \mathbb{N}$, or of every order in the case when $k=\infty$. By an open arc (a closed curve) we mean, depending on the context, a continuous injection of a finite interval (a circle of nonzero radius, respectively) in $\mathbb{C}$ or the image of such an injection. A real function is smooth if it is continuously differentiable, and smooth arcs or closed curves are ones defined by smooth injections. We denote the restriction of a function $f$ to a set $X$ by $f{\lfloor}_X$. Proposition 3.1. Let $L$ be the union of a finite number of bounded disjoint open arcs or a smooth closed curve in $\mathbb{C}$ such that for a pair of disjoint bounded domains $\{D_j\}_{j=0,1}$ with piecewise smooth boundaries in $\mathbb{C}$ this union $L$ lies in the intersection $\partial D_0\cap \partial D_1$ of their boundaries, so that the $D_j$ lie on different sides of $L$ in the following sense: at each point $z\in L$ the inward normals $\vec{\mathbf n}_{D_0} (z)$ to $D_0$ and $\vec{\mathbf n}_{D_1} (z)$ to $D_1$ have opposite directions. Let the pair of functions $V_j\in C^1(\overline D_j)\cap C^2( D_j)$ satisfy the following three conditions: If the derivative of $V_j$ with respect to the inward normal is positive outside $L$, that is,
$$
\begin{equation}
\dfrac{\partial V_j}{\partial \vec{\mathbf n}_{D_j} }\geqslant 0 \quad\textit{on } \partial D_j\setminus L \textit{ for } j=0, 1,
\end{equation}
\tag{3.2}
$$
then for each function $ U\in C^1(\overline D)\cap C^2( D) $ such that $U\leqslant 0$ on $\overline D$
$$
\begin{equation}
\iint_{D}V\Delta U\,\mathrm{d} \mathfrak{m}_2\leqslant \int_L U\biggl(\frac{\partial V_0}{\partial \vec{\mathbf n}_{D_0} }+ \frac{\partial V_1}{\partial \vec{\mathbf n}_{D_1} }\biggr)\,\mathrm{d} \mathfrak{m}_1, \quad\textit{where } \vec{\mathbf n}_{D_0} +\vec{\mathbf n}_{D_1} =\vec{0} \quad\textit{on } L.
\end{equation}
\tag{3.3}
$$
Proof. By Green’s second formula for the domains $D_j$ we have
$$
\begin{equation*}
\iint_{D_j}(V_j\Delta U-U\Delta V_j)\,\mathrm{d} \mathfrak{m}_2 \underset{j=1,2}{=}-\int_{\partial D_j} \biggl(V_j\frac{\partial U}{\partial \vec{\mathbf n}_{D_j} } -U\frac{\partial V_j}{\partial \vec{\mathbf n}_{D_j} }\biggr)\,\mathrm{d} \mathfrak{m}_1.
\end{equation*}
\notag
$$
Hence by condition (i) that $V_j$ vanishes on $ \partial D_j\setminus L$ we have
$$
\begin{equation}
\begin{aligned} \, \notag \iint_{D_j}V_j\Delta U\,\mathrm{d} \mathfrak{m}_2 &=\int_{D_j}U\Delta V_j\,\mathrm{d} \mathfrak{m}_2+ \int_{\partial D_j\setminus L}U\frac{\partial V_j}{\partial \vec{\mathbf n}_{D_j} }\,\mathrm{d} \mathfrak{m}_1 \\ &\qquad +\int_{L} \biggl(U\frac{\partial V_j}{\partial \vec{\mathbf n}_{D_j} } -V_j\frac{\partial U}{\partial \vec{\mathbf n}_{D_j} } \biggr)\,\mathrm{d} \mathfrak{m}_1 \quad \text{for } j=0,1. \end{aligned}
\end{equation}
\tag{3.4}
$$
As $U$ us negative, since $\Delta V_j\geqslant 0$, the first integral on the right-hand side of (3.4) is negative by condition (iii), and by condition (3.2) the second integral there is also negative. Hence from (3.4) we obtain
$$
\begin{equation*}
\iint_{D_j}V_j\Delta U\,\mathrm{d} \mathfrak{m}_2 \leqslant \int_{L} \biggl(U\frac{\partial V_j}{\partial \vec{\mathbf n}_{D_j} } -V_j\frac{\partial U}{\partial \vec{\mathbf n}_{D_j} } \biggr)\,\mathrm{d} \mathfrak{m}_1 \quad \text{for } j=0,1.
\end{equation*}
\notag
$$
Adding the two inequalities for $j=1$ and $j=2$, for the continuous function $V$ in (3.1) (since $V_0{\lfloor}_{L}=V_1{\lfloor}_{L}=V{\lfloor}_{L}$ by (ii)) we obtain
$$
\begin{equation*}
\iint_{D_0\cup D_1}V\Delta U\,\mathrm{d} \mathfrak{m}_2 \leqslant \int_L U\biggl(\frac{\partial V_0}{\partial \vec{\mathbf n}_{D_0} }+ \frac{\partial V_1}{\partial \vec{\mathbf n}_{D_1} }\biggr)\,\mathrm{d} \mathfrak{m}_1 -\int_L V\biggl(\frac{\partial U}{\partial \vec{\mathbf n}_{D_0} }+ \frac{\partial U}{\partial \vec{\mathbf n}_{D_1} }\biggr) \,\mathrm{d} \mathfrak{m}_1.
\end{equation*}
\notag
$$
Here the second integral on the right vanishes because the inward normals have opposite directions, so that $\vec{\mathbf n}_{D_0} +\vec{\mathbf n}_{D_1} =\vec{0}$ on $L$, and the integral over $D_0\cup D_1$ on the left can be replaced by the integral over $D=D_0\cup D_1\cup L$ because ${\mathfrak{m}_2}(L)=0$, which proves (3.3).
Proposition 3.1 is proved. We say that a function $u\colon S\to \overline{\mathbb{R}}$ is subharmonic on a set $S\subset \mathbb{C}$ if there exists a subharmonic function on a neighbourhood of $S$ whose restriction to $S$ coincides with $u$. The difference $U=u-v$ of two subharmonic functions $u\neq -\infty$ and $v\neq -\infty$ on a bounded connected set $S\subset \mathbb{C}$ is defined $\mathfrak{m}_1$-almost everywhere, and it is always $\mathfrak{m}_1$-summable on each Lipschitz arc whose image lies in $S$. For such a difference $U=u-v$ the Riesz charge distribution $\varDelta_U:=\varDelta_u-\varDelta_v$ is well defined. Given a Borel measure $\mu$ on a Borel set $S\subset \mathbb{C}$, if for a subset $E$ of $ S$ of measure $\mu(E)=0$ an extended real function $U\colon S\setminus E\to \overline{\mathbb{R}}$ is negative, then we say that $U$ is negative $\mu$-almost everywhere on $S$. If the difference $U=u-v$ of two subharmonic functions $u\neq -\infty$ and $v\neq -\infty$ is negative $\mathfrak{m}_2$-almost everywhere in the domain, then it is also negative $\mathfrak{m}_1$-almost everywhere in this domain, and moreover, $u\leqslant v$ everywhere in the domain. Corollary 3.1. Under the assumptions of Proposition 3.1 up to the definition (3.1) of the function $V$ inclusive, let $U$ be a difference of two subharmonic functions distinct from $-\infty$ on $\overline D$, with Riesz mass distribution $\varDelta_U$. If each function $V_j$ is positive on $\overline D_j$, and the difference $U$ is negative $\mathfrak{m}_2$-almost everywhere in a neighbourhood of the closure $\overline D$, then
$$
\begin{equation}
\iint_{D}V\,\mathrm{d}\varDelta_U\leqslant \frac{1}{2\pi}\int_L U\biggl(\frac{\partial V_0}{\partial \vec{\mathbf n}_{D_0} }+ \frac{\partial V_1}{\partial \vec{\mathbf n}_{D_1} }\biggr)\,\mathrm{d} \mathfrak{m}_1, \quad\textit{where } \vec{\mathbf n}_{D_0} +\vec{\mathbf n}_{D_1} =\vec{0} \quad\textit{on } L.
\end{equation}
\tag{3.5}
$$
Proof. Since the functions $V_j$ are positive on $\overline D_j$ and vanish on $\partial D_j\setminus L$, by the definition of the derivative along to the inward normal, their inward normal derivatives on $\partial D_j\setminus L$ are positive, so that condition (3.2) in Proposition 3.1 is satisfied. Thus, if $U\leqslant 0$ is a difference of twice continuously differentiable subharmonic functions, then (3.5) is just inequality (3.3) in Proposition 3.1, because $\varDelta_U=\frac1{2\pi}\Delta U$.
Now let $U=u-v$ be a difference of the subharmonic functions $u\neq -\infty$ and $v\neq -\infty$ in a neighbourhood of the compact set $\overline D$, and provisionally let $u\in C^2(\overline D)$. Then there exist a domain $D'$ containing $\overline D$ and a decreasing sequence $(v_k)_{k\in \mathbb{N}}$ of infinitely differentiable subharmonic functions in $D'$ that tends to $v$ pointwise (see [8], Theorem 3.8, and [27], Theorem 2.7.2), while the sequence of their Riesz mass distributions $(\varDelta_{v_k})_{k\in \mathbb{N}}$ converges weakly at continuous functions with compact support in $D'$ to the Riesz mass distribution $\varDelta_v$ (see [25], Ch. III, § 3, Theorem 3.9). In particular, the integrals with respect to $\varDelta_{v_k}$ of the function $V$ in (3.1), which is by definition continuous and has a compact support in $D'$, tend to the integrals of $V$ with respect to $\varDelta_{v}$. Since the sequence $(v_k)_{k\in \mathbb{N}}$ decreases, for each $k$ we still have $U_k=u-v_k\leqslant 0$ $\mathfrak{m}_2$-almost everywhere in $D'$, and therefore everywhere in $D'$. By the above (3.5) holds for each difference $U_k:=u-v_k$ as $U$, with respect to the charge distribution $\varDelta_u-\varDelta_{v_k}$. Hence, taking first the limit as $k\to +\infty$, we also obtain the required inequality (3.5) for such differences $U=u-v$. It remains to drop the condition $u\in C^2(\overline D)$. Now let $U=u-v$ be a difference of arbitrary subharmonic functions $u\neq -\infty$ and $v\neq -\infty$ in a neighbourhood of the compact set $\overline D$. Fixing some $\varepsilon >0$, consider the difference $U_\varepsilon :=(u-\varepsilon) -v$ of the subharmonic functions $u-\varepsilon$ and $v$. The scheme of construction of a sequence of infinitely differentiable subharmonic functions decreasing to a prescribed subharmonic function consists in taking the averaging convolutions over shrinking discs with infinitely differentiable positive functions with support in these discs (see [8], Theorem 3.8, and [27], Theorem 2.7.2). On the basis of this scheme we can construct a sequence $(u_n)_{n\in \mathbb{N}}$ of subharmonic functions $u_n\in C^{\infty}(D')$ decreasing to the upper semicontinuous function $u\varepsilon$ on some fixed domain $D'\supset \overline D$ so that we still have $u_n-v\leqslant 0$ for all $n\in \mathbb{N}$ in a neighbourhood of $\overline D$. It follows from the above that inequality (3.5) holds for the differences $u_n-v\leqslant 0$. Taking the limit as $n\to \infty$ and using the same arguments for the Riesz mass distributions $\varDelta_{u_n}$, converging weakly to $\varDelta_{u-\varepsilon}=\varDelta_{u}$, we obtain inequality (3.5) for the function $U_\varepsilon$ as $U$. In this inequality $\varepsilon$ is only involved in the integrand on the right-hand side of (3.5), in the function $U-\varepsilon$ replacing $U$. Then we let $\varepsilon$ tend to zero on the right-hand side of the inequality, which yields (3.5) without the assumption that the difference $U$ is twice continuously differentiable. Corollary 3.1 is proved. § 4. Some classes of generalized convex functionsIn this section we only consider directly those simple properties of $\log$-convex and $ p$-power convex functions that we require in what follows. 4.1. Representations in terms of convex functionsProposition 4.1. Let $F\colon I\to \mathbb{R}$ be a function on an interval $I\subset \mathbb{R}^+$. Then the following three assertions are equivalent: The function $F$ in (i) is decreasing if and only if so is the convex function $g$ in (ii) and (iii). Proof. By the definition of $\log$-convex functions $F$ in § 1.2.5 such functions are characterized by implications involving inequalities of the unified form $F(t)\leqslant c_1\log t+c_2$ for $t\in I$, where $c_1,c_2\in \mathbb{R}$. The substitution $x:=\log t$ reduces such an inequality to the form $g(x)\underset{x\in \log I}{=}F(e^x)\leqslant c_1x+c_2$. And such inequalities, by the definition in § 1.2.5, ensure the convexity of the function $g$ on $\log I$. This proves the equivalence (i) $\Leftrightarrow$ (ii), while the equivalence (ii) $\Leftrightarrow$ (iii) is ensured by the form of $g$ in (ii) and the inverse substitution $t:=e^x$. The addendum concerning the decrease is obvious.
The proof is complete. Proposition 4.2. Let $F\colon I\to \mathbb{R}$ be a function on an interval $I\subset \mathbb{R}^+$, and let $0<p\in \mathbb{R}^+$. Then the following three assertions are equivalent: Proof. By the definition of $ p$-power convex functions $F$ in § 1.2.5, they are fully characterized by implications involving inequalities of the form $F(t)\leqslant c_1t^p+c_2t^{-p}$ for $t\in I$, with $c_1,c_2\in \mathbb{R}$. These inequalities can equivalently be written as $t^pF(t)\leqslant c_1t^{2p}+c_2$ for $t\in I$. Making the substitution $x:=t^{2p}$ we transform this relation into
$$
\begin{equation*}
g(x)\underset{x\in I^{2p}}{=}\sqrt{x}F(x^{1/(2p)})\leqslant c_1x+c_2.
\end{equation*}
\notag
$$
By the definition in § 1.2.5 such inequalities characterize the convexity of $g$ on $I^{2p}$. This proves the equivalence (i) $\Leftrightarrow$ (ii), while the equivalence (ii) $\Leftrightarrow$ (iii) is ensured by the form of $g$ in (4.2) and the inverse substitution $t:=x^{1/(2p)}$.
The proof is complete. From Propositions 4.1 and 4.2, representations (4.1) and (4.3) there and some well-known simple properties of ordinary convex functions we obtain the following result. Proposition 4.3. The classes of $\log$-convex and $ p$-power convex functions on an interval $I\subset \mathbb{R}^+$ are closed under multiplication by a positive constant and addition. 4.2. Continuity, constants, decrease and differentiationFrom representations (4.1) and (4.3) in Propositions 4.1 and 4.2 and well-known simple properties of ordinary convex functions we can deduce the following result. Proposition 4.4. For $p\in \mathbb{R}^+$ each $p$-power convex function on an interval $I\subset \mathbb{R}^+$ is continuous on $I$ and has a right and a left derivative everywhere on $I$, which coincide away from an at most countable subset, so that the function is differentiable away from this subset, and these derivatives are Riemann integrable on each closed subinterval of $I$. Proposition 4.5. The class of $\log$-convex function on an interval $I\subset \mathbb{R}^+$ contains all constants. For $p>0$ a constant is $ p$-power convex if and only if it is negative. Proof. The first result is obvious. Let $p>0$. For each constant $c\in \mathbb{R}$ the function $q$ in assertion (i) of Proposition 4.2 can be defined by
$$
\begin{equation*}
g(x)\overset{(4.2)}{\underset{x\in I^{2p}}{=}}\sqrt{x}\cdot c.
\end{equation*}
\notag
$$
This function is convex only for $c\leqslant 0$. It remains to use the equivalence (i)$\Leftrightarrow$(ii) from Proposition 4.2.
The proof is complete. Proposition 4.6. Let $F\colon (r,R)\to \mathbb{R}^+$ be a decreasing $\log$-convex positive function on the interval $(r,R)\subset \mathbb{R}^+$, where $r>0$, and let and Then there exists a decreasing convex positive function F:
$$
\begin{equation}
g\colon [\log r,\log R]\to \mathbb{R}^+, \qquad g(\log R)\overset{(4.4)}{:=}0, \qquad g(\log r)\overset{(4.5)}{:=}F(r)-F(R),
\end{equation}
\tag{4.6}
$$
that represents the $\log$-convex positive decreasing function Moreover, at $R$ this function has a finite negative left derivative
$$
\begin{equation}
F'_{\mathrm{lh}}(R):=\lim_{R>t\to R}\frac{F(t)-F(R)}{t-R}\in -\mathbb{R}^+,
\end{equation}
\tag{4.8}
$$
and at $r$ it has a negative right derivative
$$
\begin{equation}
F'_{\mathrm{rh}}(r):=\lim_{r<t\to r}\frac{F(t)-F(r)}{t-r}\in -\overline{\mathbb{R}}^+
\end{equation}
\tag{4.9}
$$
which can be equal to $-\infty$. Thus, there exist left-continuous left derivatives and, provided that $F'_{\mathrm{rh}}(r)>-\infty$, also right-continuous right derivatives We leave out the elementary proof of this proposition. Proposition 4.7. Let $0<p\in \mathbb{R}^+$, and let $F\colon (r,R)\to \mathbb{R}^+$ be a decreasing $ p$-power convex positive function on $(r,R)\subset \mathbb{R}^+$, where $r>0$ and relations (4.4) and (4.5) hold. Then there exists a decreasing convex positive function
$$
\begin{equation}
g\colon (r^{2p},R^{2p})\to \mathbb{R}^+, \qquad g(R^{2p})\overset{(4.4)}{:=}0, \qquad g(r^{2p})\overset{(4.5)}{:=}r^p\bigl(F(r)-F(R)\bigr),
\end{equation}
\tag{4.12}
$$
which represent the decreasing positive $p$-power convex function $F$:
$$
\begin{equation}
F(t)-F(R)\underset{t\in [r,R]}{=}\frac{g(t^{2p})}{t^p}.
\end{equation}
\tag{4.13}
$$
Moreover, $F$ has a finite negative left derivative (4.8) at $R$, and a right derivative (4.9), maybe equal to $-\infty$, at $r$. Thus, there exist left continuous left derivatives and, provided that $F'_{\mathrm{rh}}(r)>-\infty$, also right continuous right derivatives Proof. It follows from Propositions 4.5 and 4.3 that $F-F(R)$ is $p$-power convex on $(r,R)$ as the sum of $F$ and the negative constant $-F(R)$. It is obvious from (4.12) that the convex function $g$ that is defined on $I^{2p}=(r^{2p},R^{2p})$ (where $I:=(r,R)$) for the function $F-F(R)$ in (4.2) by assertion (ii) of Proposition 4.2 and is involved in representation (4.13) on $[r,R]$ is positive, and it is continuous on $(r^{2p}, R^{2p})$ because it is convex. That $g$ is left continuous at $R^{2p}$ and right continuous at $r^{2p}$ follows from (4.4) and (4.5) in view of (4.12). In addition, $g(\mathbb{R}^{2p})\overset{(4.12)}{=}0$, and $g$ is positive and convex, so it is decreasing.
We mentioned in Proposition 4.4 that $F$ has left and right derivatives on $(r,R)$. It follows directly from (4.13) that these have the form (4.14) and (4.15) on $(r,R)$. As $g$ is a decreasing convex positive function, left continuous at $R^{2p}$ and vanishing at these points, all left derivatives increase to the left of $R^{2p}$, but remain negative because $\bigl(g(t)-g(R^{2p})\bigr)/(t-r)\leqslant 0$ for $t\leqslant R^{2p}$. Hence the limit of left derivatives $\lim_{R^{2p}>t\to R^{2p}}g'_{\mathrm{lh}}(t)$ exists and is negative, so that
$$
\begin{equation*}
g'_{\mathrm{lh}}(R^{2p})\overset{(4.14)}{=} \frac{1}{2R^{2p}}\biggl(\frac{R^{p+1}}{p}F'_{\mathrm{lh}}(R)+g(R^{2p})\biggr) \overset{(4.12)}{=}\frac{1}{2pR^{p-1}}F'_{\mathrm{lh}}(R).
\end{equation*}
\notag
$$
Hence we can extend equality (4.14) from $t<R$ to $t=R$. By the finiteness condition $F'_{\mathrm{rh}}(r)>-\infty$ and (4.15) we have
$$
\begin{equation*}
g'_{\mathrm{rh}}(r^{2p})\overset{(4.15)}{=} \frac{1}{2r^{2p}}\biggl(\frac{r^{p+1}}{p}F'_{\mathrm{rh}}(t)+g(r^{2p})\biggr) =\frac{1}{2r^{p}}\biggl(\frac{r}{p}F'_{\mathrm{rh}}(t)+F(r)\biggr),
\end{equation*}
\notag
$$
so we can extend equality (4.15) from $t>r$ to $t= r$. The left (right) derivatives in (4.14) (in (4.15), respectively) are left (right) continuous by the well-known properties of convex functions as applied to $g$.
Proposition 4.7 is proved. Proposition 4.8. Let $0<R\leqslant +\infty$, and let $F$ be a decreasing positive $\log$-convex function on an interval $(0,R)$. If then Proof. For fixed $t\in (0,R)$ consider an arbitrary point $r\in (0,t)$. By Proposition 4.6 some decreasing convex positive function $g$ on $I:=[\log r,\log R)$ represents ${F-F(R)}=g\circ\log$ as in (4.6) and (4.7). The following result is a consequence of simple geometric relations for the slopes of half-tangents and secants.
Lemma 4.1. If $g\colon I\to \mathbb{R}$ is a decreasing convex function on an interval $I\subset \mathbb{R}$, then at each interior point $x$ of $I$ From this lemma we obtain
$$
\begin{equation*}
0\geqslant g'_{\mathrm{rh}}(\log t)\overset{(4.18)}{\geqslant} \frac{g(\log r)-g(\log t)}{\log (r/t)} \overset{(4.7)}{=}\frac{F(r)-F(t)}{\log (r/t)}.
\end{equation*}
\notag
$$
Hence, using equality (4.11) in Proposition 4.6 we obtain
$$
\begin{equation*}
\label{F*r00} 0\geqslant F'_{\mathrm{rh}}(t)\overset{(4.11)}{=} \frac{1}{t}g'_{\mathrm{rh}}(\log t)\geqslant \frac{1}{t}\frac{F(r)-F(t)}{\log (r/t)}.
\end{equation*}
\notag
$$
Here we can take the lower limit as $r\to 0$ on the right-hand side, which yields
$$
\begin{equation*}
0\geqslant F'_{\mathrm{rh}}(t) \geqslant \liminf_{0<r\to 0} \frac{1}{t}\frac{F(r)-F(t)}{\log (r/t)} = \frac{1}{t}\liminf_{0<r\to 0} \frac{F(r)}{\log (r/t)} \overset{(4.16)}{=} \frac{1}{t}\bigl(-{\lim}_0^0F\bigr).
\end{equation*}
\notag
$$
This completes the proof of (4.17). Proposition 4.8 is proved. Proposition 4.9. Let $0<R\in \mathbb{R}^+$ and $0<p\in \mathbb{R}^+$, and let $F\colon (0,R)\to \mathbb{R}^+$ be a decreasing $p$-power convex positive function on $(0,R)$. If then Proof. Fix $t\in (0,R)$, and let $r\in (0,t)$. By Proposition 4.7, for $p>0$ there exists a decreasing positive continuous function $g$ on $I:=[r^{2p}, R^{2p})$ that represents $F-F(R)$ as in (4.12) and (4.13). From Lemma 4.1 and (4.18) we obtain
$$
\begin{equation*}
\begin{aligned} \, 0 &\geqslant g'_{\mathrm{rh}}(t^{2p})\overset{(4.18)}{\geqslant} \frac{g(r^{2p})-g(t^{2p})}{r^{2p}-t^{2p}} \\ &\!\!\!\!\overset{(4.13)}{=}\frac{r^{p}\bigl(F(r)-F(R)\bigr) -t^p\bigl(F(t)-F(R)\bigr)}{r^{2p}-t^{2p}} =\frac{r^{p}F(r)-t^pF(t)}{r^{2p}-t^{2p}} -\frac{F(R)}{r^{p}+t^{p}}. \end{aligned}
\end{equation*}
\notag
$$
Here we can take the lower limit as $0<r\to 0$ on the right-hand side, so that
$$
\begin{equation*}
\begin{aligned} \, 0 &\geqslant g'_{\mathrm{rh}}(t^{2p})\geqslant \frac{1}{t^{2p}}\liminf_{0<r\to 0}\bigl(-r^pF(r)\bigr) +\lim_{0<r\to 0} \biggl(\frac{-t^pF(t)}{r^{2p}-t^{2p}} -\frac{F(R)}{r^{p}+t^{p}}\biggr) \\ &\!\!\!\!\overset{(4.19)}{=} -\frac{{\lim}_0^pF}{t^{2p}} +\frac{F(t)-F(R)}{t^p}. \end{aligned}
\end{equation*}
\notag
$$
Using this inequality for $p>0$, by positivity (4.15) in Proposition 4.7 and representation (4.13) we obtain
$$
\begin{equation*}
\begin{aligned} \, 0 &\geqslant F'_{\mathrm{rh}}(t)\overset{(4.15)}{=}\frac{p}{t^{p+1}}\bigl(2g'_{\mathrm{rh}} (t^{2p})t^{2p}-g(t^{2p})\bigr) \overset{(4.13)}{=}\frac{p}{t}\bigl(2g'_{\mathrm{rh}} (t^{2p})t^{p}-\bigl(F(t)-F(R)\bigr)\bigr) \\ &\geqslant \frac{p}{t}\biggl(2\biggl(-\frac{{\lim}_0^pF}{t^{2p}} +\frac{F(t)-F(R)}{t^p}\biggr)t^{p}-\bigl(F(t)-F(R)\bigr)\biggr) \\ & =-2p\frac{{\lim}_0^pF}{t^{p+1}}+p\frac{F(t)-F(R)}{t}. \end{aligned}
\end{equation*}
\notag
$$
Hence subtraction of the last term and multiplication by $t$ yield
$$
\begin{equation*}
-p\bigl(F(t)-F(R)\bigr)\geqslant tF'_{\mathrm{rh}}(t)-p\bigl(F(t)-F(R)\bigr)\geqslant -2p\frac{{\lim}_0^pF}{t^p}.
\end{equation*}
\notag
$$
The middle expression in this chain is the opposite of $Q_{p,F}(t)$ in (4.20). Multiplying by $-1$ and reversing the inequality signs we obtain (4.20).
Proposition 4.9 is proved. 4.3. Extension and approximationIn what follows we will twice require Proposition 4.10 below on the affine extension and approximation of ordinary convex functions by infinitely differentiable ones on intervals. Proposition 4.10. Let $g\colon (a,b)\to \mathbb{R}$ be a convex function on a finite interval $(a,b)\subset \mathbb{R}$, and let there exist finite limits Then the following assertions hold. (i) The affine extension of $g$ to $\mathbb{R}$ defined by
$$
\begin{equation}
G\colon x\underset{x\in \mathbb{R}}{\longmapsto} \begin{cases} g(a)+g_{\mathrm{rh}}(a)(x-a)&\textit{for } x\in (-\infty,a), \\ g(x)\quad&\textit{for } x\in [a,b], \\ g(b)+g_{\mathrm{lh}}(b)(x-b)&\textit{for } x\in (b,+\infty), \end{cases}
\end{equation}
\tag{4.23}
$$
is a convex function on $\mathbb{R}$ which is differentiable at $a$ and $b$, and
$$
\begin{equation}
G'(x)= \begin{cases} g_{\mathrm{rh}}(a)&\textit{for } x\in (-\infty,a], \\ g_{\mathrm{lh}}(b)&\textit{for } x\in [b,+\infty). \end{cases}
\end{equation}
\tag{4.24}
$$
If $g$ is decreasing or increasing on $[a,b]$, then $G$ in (4.23) is also decreasing or increasing on $[a,b]$. (ii) For the affine extension $G$ in (4.23) there exists a decreasing sequence $(G_n)$ of infinitely differentiable convex functions $G_n$ on $\mathbb{R}$ that converges to $G$ uniformly on $\mathbb{R}$, and such that
$$
\begin{equation}
G_n\overset{(4.23)}{=}G \quad\textit{on } \mathbb{R}\setminus \biggl(a-\frac1n,b+\frac1n\biggr)
\end{equation}
\tag{4.25}
$$
and (4.24) holds for $a-1/n$ in place of $a$ and $b+1/n$ in place of $b$. If $g$ is decreasing or increasing on $[a,b]$, then each $G_n$ is decreasing or increasing on $\mathbb{R}$. In addition, Proof. Assertion (i) follows directly from (4.23). For the construction of $(G_n)_{n\in \mathbb{N}}$ in (ii) we can use the standard convolutions $G_n:=G*k_n$ of the affine extension $G$ from (4.23) with the functions $ k_n(t)\underset{t\in \mathbb{R}}{:=}k(nt)n$, $n\in \mathbb{N}$, where $k\geqslant 0$ is an infinitely differentiable even function on $\mathbb{R}$ with compact support and $\displaystyle \int k(t)\,\mathrm{d} t=1$. This yields immediately all the required properties of the sequence $G_n:=G*k_n$, apart from uniform convergence. A sequence of convex functions converging pointwise to a real function on an interval converges uniformly on compact subintervals of this interval (see [22], Theorem 13E). In particular, $G_n$ converges uniformly to $G$ on $[a-1,b+1]$. On the complement $\mathbb{R}\setminus [a-1, b+1]$ all functions $G_n$ are equal to $G$. Hence we have uniform convergence on $\mathbb{R}$. To prove the limit relation (4.26) for derivatives we can use a rather general result (see [38], Theorem 7) and some details of its proof. By this result, given a pointwise convergent sequence of convex continuously differentiable functions on an interval, at each point of differentiability of the limit function the sequence or derivatives converges to the derivative of the limit function at this point. On the other hand, for a decreasing sequence $G_n$ our argument can be more geometric. The fact that $G$ is differentiable at a point $a$ means that the convex epigraph of $G$ has a unique support line $l$ at $a$. The epigraph of $G$ coincides with the union of increasing epigraphs of the decreasing sequence of functions $G_n$, which also have unique support lines $l_n$ at $a$. Clearly, the sequence of these lines $l_n$ converges to $l$ in a certain sense. In the language of derivatives this is relation (4.26).
Proposition 4.10 is proved. Proposition 4.11. Let $F\colon (r,R)\to \mathbb{R}^+$ be a decreasing positive $\log$-convex function on an interval $(r,R)\subset \mathbb{R}^+$, $r>0$, with values (4.4) and (4.5) at the endpoints, and assume that the right derivative $F'_{\mathrm{rh}}(r)>-\infty$ at the left-hand endpoint $r$ is finite. Then the difference $F-F(R)$ can be extended to $\mathbb{R}^+\setminus \{0\}$ as a decreasing $\log$-convex function with derivative continuous outside the interval $(r,R)$. Moreover, there exists a sequence $(F_n)_{n\in \mathbb{N}}$ of infinitely differentiable decreasing $\log$-convex functions $F_n$ on $\mathbb{R}^+\setminus \{0\}$ satisfying $F_n(R)\underset{n\in \mathbb{N}}{=}0$, converging uniformly to $F-F(R)$ on $[r,R]$ and such that at $r$. Proof. The assumptions on $F$ are here the same as in Proposition 4.6. By construction and by the properties of the convex function $g$ in Proposition 4.6 that represents $F-F(R)$ in (4.7), at the endpoints of the interval $(a,b):=(\log r,\log R)$, the assumptions (4.21) and (4.22) of Proposition 4.10 are fulfilled by (4.6) and (4.10), (4.11). We define an extension of the function $F-F(R)$ to $\mathbb{R}^+\setminus \{0\}$ in terms of the affine extension $G$ of $g$ to $\mathbb{R}$ defined in assertion (i) of Proposition 4.10, as a continuous function
$$
\begin{equation}
t\overset{(4.23),(4.1)}{\underset{0<t\in \mathbb{R}^+}{\longmapsto}}G(\log t) \quad\text{by setting } F(t)-F(R)\underset{t\in [r,R]}{=}G(\log t).
\end{equation}
\tag{4.27}
$$
Since $G$ is convex on $\mathbb{R}^+\setminus \{0\}$, by the equivalence (i) $\Leftrightarrow$ (iii) from Proposition 4.1 the function (4.27) is by construction a $\log$-convex function, and it is continuously differentiable on $\bigl(\mathbb{R}^+\setminus \{0\}\bigr)\setminus (r,R)$ in view of (4.24).
Let $(G_n)_{n\in \mathbb{N}}$ be a decreasing sequence of infinitely differentiable decreasing convex functions $G_n$ on $\mathbb{R}$ that converges to $G$ uniformly on $\mathbb{R}$ and has the properties (4.25), (4.24) and (4.26) for $a:=\log r$ and $b:=\log R$. Set
$$
\begin{equation}
F_n\colon t \underset{t\in \mathbb{R}^+\setminus \{0\}}{\longmapsto} G_n(\log t)-G_n(\log R).
\end{equation}
\tag{4.28}
$$
By construction the $F_n$ are infinitely differentiable and decreasing on $\mathbb{R}^+\setminus \{0\}$, and the sequence $(F_n)_{n\in \mathbb{N}}$ converges to $F-F(R)$ uniformly on $[r,R]$ because of the representation
$$
\begin{equation*}
F(t)-F(R)\underset{t\in [r,R]}{=}g(\log t)\underset{t\in [r,R]}{=}G(\log t)
\end{equation*}
\notag
$$
and in view of the limit relation
$$
\begin{equation*}
G_n(\log R)\underset{n\to \infty}{\longrightarrow} g(\log R)=0;
\end{equation*}
\notag
$$
in addition, $F_n(R)\overset{(4.28)}{=}0$. By Propositions 4.1, 4.5 and 4.3 all functions $F_n$ are convex with respect to $\log$, and the limit relation
$$
\begin{equation*}
\lim_{n\to \infty}F_n'(r)\overset{(4.9)}{=}F'_{\mathrm{rh}}(r)
\end{equation*}
\notag
$$
for derivatives at $r$ is a consequence of 4.26 for $a:=\log r$, representations (4.27) and (4.28) for the functions $F-F(R)$ and $F_n$, and the method of calculation of the right derivative (4.11) of $F$.
Proposition 4.11 is proved. Proposition 4.12. Let $0<p\in \mathbb{R}^+$, and let $F\colon (r,R)\to \mathbb{R}^+$ be a decreasing $ p$-power convex positive function on an interval $(r,R)\subset \mathbb{R}^+$, where $r>0$, and assume that relations (4.4) and (4.5) hold and the right derivative $F'_{\mathrm{rh}}(r)>-\infty$ at the left-hand endpoint $r$ is finite. Then $F-F(R)$ can be extended to $\mathbb{R}^+\setminus \{0\}$ as a decreasing $ p$-power convex function whose derivative is continuous outside $ (r,R)$. Furthermore, there exists a sequence of infinitely differentiable decreasing $p$-power convex functions $F_n$ on $\mathbb{R}^+\setminus \{0\}$ that converge uniformly to $F-F(R)$ on $[r,R]$ and such that $F_n(R)\underset{n\in \mathbb{N}}{=}0$ and at $r$. Proof. The assumptions on $F$ are here the same as in Proposition 4.7. By construction and by the properties of the convex function $g$ in Proposition 4.7 which represents $F-F(R)$ in (4.13), assumptions (4.21) and (4.22) of Proposition 4.10 are fulfilled at the endpoints of the interval $(a,b):=(r^{2p},R^{2p})$, according to (4.12) and (4.14), (4.15).
We can extend $F-F(R)$ to $\mathbb{R}^+\setminus \{0\}$ in terms of the affine extension $G$ of $g$ to $\mathbb{R}$ defined in assertion (i) of Proposition 4.10, as a decreasing function
$$
\begin{equation}
t\overset{(4.23),(4.1)}{\underset{0<t\in \mathbb{R}^+}{\longmapsto}}\frac{G(t^{2p})}{t^p} \quad\text{by setting } F(t)-F(R)\underset{t\in [r,R]}{=}\frac{G(t^{2p})}{t^p}.
\end{equation}
\tag{4.29}
$$
Since $G$ is convex $G$ on $\mathbb{R}^+\setminus \{0\}$, by the equivalence (i) $\Leftrightarrow$ (iii) in Proposition 4.2 the function (4.29) is by construction a $p$-power convex function, which is continuously differentiable outside $(r,R)$ by (4.24).
Let $(G_n)_{n\in \mathbb{N}}$ be a decreasing sequence of infinitely differentiable decreasing convex functions $G_n$ on $\mathbb{R}$ that converges to $G$ uniformly on $\mathbb{R}$, and has the properties (4.25), (4.24) and (4.26) for $a:=r^{2p}$ and $b:=R^{2p}$. Set
$$
\begin{equation}
F_n\colon t \underset{t\in \mathbb{R}^+\setminus \{0\}}{\longmapsto} \frac{G_n(t^{2p})}{t^p}-\frac{G_n(R^{2p})}{R^p}.
\end{equation}
\tag{4.30}
$$
By construction the $F_n$ are infinitely differentiable and decreasing on $\mathbb{R}^+\setminus \{0\}$, and the sequence $(F_n)_{n\in \mathbb{N}}$ converges to $F-F(R)$ uniformly on $[r,R]$ by representation (4.29) and since
$$
\begin{equation*}
\lim_{n\to \infty}\frac{G_n(R^{2p})}{R^p}=\frac{g(R^{2p})}{R^p}\overset{(4.12)}{=}0;
\end{equation*}
\notag
$$
in addition, $F_n(R)\overset{(4.30)}{=}0$. By Propositions 4.2, 4.5 and 4.3 all functions $F_n$ are $ p$-power convex, and the relation $\lim_{n\to \infty}F_n'(r)\overset{(4.9)}{=}F'_{\mathrm{rh}}(r)$ at $r$ follows from (4.26) for $a:=r^{2p}$, representations (4.29) and (4.30) for $F-F(R)$ and $F_n$, and the method of calculation of the right derivative (4.15) of $F$.
Proposition 4.12 is proved. 4.4. Functions with separated polar variablesProposition 4.13. Let $p\in \mathbb{R}^+$, let $s\colon\mathbb{R}\to \mathbb{R}^+$ be a twice continuously differentiable $2\pi$-periodic positive $p$-trigonometrically convex function, let ${0\,{<}\,r\,{<}\,R\,{<}\,+\infty}$, and let $F\colon (r,R)\to \mathbb{R}^+$ be a twice continuously differentiable decreasing positive function that is $ p$-power convex, so that it is $\log$-convex for $p=0$. Then for $F(R):=\lim_{R>t\to R}F(t)$, provided that $F(r):=\lim_{r<t\to r}F(t)<+\infty$ and $F'_{\mathrm{rh}}(r)> -\infty$, the function
$$
\begin{equation}
V\colon te^{i\theta}\underset{r\leqslant t\leqslant R,\, \theta\in \mathbb{R}}{\longmapsto} s(\theta)\bigl(F(t)-F(R)\bigr)\in \mathbb{R}^+
\end{equation}
\tag{4.31}
$$
is positive, belongs to the class $C^1(\overline D(R)\setminus D(r))\cap C^2(D(R)\setminus \overline D(r))$ and Proof. It is obvious from the construction (4.31) of $V$ that the function is positive and (4.32) holds, because $s$ is positive, $F$ is decreasing and by the definition of $F(R)$. That $V$ belongs to $C^2(D(R)\setminus \overline D(r))$ follows directly from the facts that $s$ is twice continuously differentiable on $\mathbb{R}$ and $F$ is so on $(r,R)$. The latter, in combination with the continuous derivatives of the extension of $F$ outside $(r,R)$, which were noted in Propositions 4.11 and 4.12, shows also that $V$ belongs to $C^1\bigl(\overline D(R)\setminus D(r)\bigr)$.
It remains to show that the Laplace operator of V is positive as in (4.33). First consider $p=0$. Then the $2\pi$-periodic positive $0$-trigonometrically convex function $s$ is constant, and the function $F-F(R)$ admits by Proposition 4.11 the representation
$$
\begin{equation*}
F(t)-F(R)\overset{(4.7)}{\underset{t\in (r,R)}{=}}g(\log t)
\end{equation*}
\notag
$$
with a convex function $g$ on $(r,R)$. Applying to $V$ in (4.31) the Laplace operator in the polar coordinates
$$
\begin{equation}
\Delta=\frac{\partial^2}{\partial t^2}+\frac1{t} \, \frac{\partial}{\partial t} +\frac{1}{t^2} \, \frac{\partial^2}{\partial\theta^2},
\end{equation}
\tag{4.34}
$$
from (4.10) and (4.11) we easily obtain
$$
\begin{equation*}
(\Delta V)(te^{i\theta})\overset{(4.34)}{\underset{t\in (r,R)}{=}}s \, \frac{\,\mathrm{d}}{\,\mathrm{d} t}\biggl(\frac{1}{t} \, g'(\log t)\biggr) +s \, \frac{1}{t}\, \frac{1}{t} \, g'(\log t)\underset{t\in (r,R)}{=}s \, \frac{1}{t^2} \, g''(\log t)\underset{t\in (r,R)}{\geqslant} 0
\end{equation*}
\notag
$$
because $g$ is convex. This proves (4.33) for $p:=0$.
Now let $p>0$. We use the representation
$$
\begin{equation*}
F(t)-F(R)\overset{(4.13)}{\underset{t\in [r,R]}{=}}\frac{g(t^{2p})}{t^p}
\end{equation*}
\notag
$$
from Proposition 4.12, where $g$ is a positive convex function. By
$$
\begin{equation}
\frac{\partial V}{\partial t} (te^{i\theta})\overset{(4.14),\,(4.15)}{=} ps(\theta)\bigl(2g'(t^{2p})t^{p-1}-g(t^{2p})t^{-p-1}\bigr)
\end{equation}
\tag{4.35}
$$
we can easily calculate the second radial derivative:
$$
\begin{equation*}
\frac{\partial^2 V}{\partial t^2} (te^{i\theta})=ps(\theta)\bigl(4pg''(t^{2p})t^{3p-2} -2g'(t^{2p})t^{p-2}+g(t^{2p})(p+1)t^{-p-2}\bigr).
\end{equation*}
\notag
$$
Now calculating the value of the Laplace operator in the polar coordinates we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \bigl(\Delta V)(te^{i\theta}) &= ps(\theta)\bigl(4pg''(t^{2p})t^{3p-2} -2g'(t^{2p})t^{p-2}+g(t^{2p})(p+1)t^{-p-2}\bigr) \\ \notag &\qquad+\frac{1}{t}ps(\theta)\bigl(2g'(t^{2p})t^{p-1}-g(t^{2p})t^{-p-1}\bigr) +\frac{1}{t^2}s''(\theta)\frac{g(t^{2p})}{t^p} \\ &=4p^2s(\theta)g''(t^{2p})t^{3p-2}+\bigl(s''(\theta)+p^2s(\theta)\bigr) \frac{g(t^{2p})}{t^{p+2}}. \end{aligned}
\end{equation}
\tag{4.36}
$$
Since $g$ is convex and $g''\geqslant 0$ and since $s$ is positive, the first term on the right in (4.36) is positive. A twice differentiable $p$-trigonometrically convex function satisfies $s''+p^2s\geqslant 0$ (see [1], Ch. I, § 16, paragraph (g), [10], Theorem 26, and [11], Theorem 7.15), and since $g$ is positive, the second term on the right in (4.36) is too. Thus, we have established the positivity requirement (4.33) for all $p>0$.
Proposition 4.13 is proved. § 5. Main results of Riesz mass and charge distributions5.1. The main inequality for differences of subharmonic functions on a disc and the planeFor an $\mathfrak{m}_1$-integrable function $M\colon \partial \overline D(r)\to \overline{\mathbb{R}}$ on the circle $\partial \overline D(r)$ its integral mean
$$
\begin{equation}
M^{\circ r}:=\frac{1}{2\pi r}\int_{\partial \overline D(r)} M\,\mathrm{d} \mathfrak{m}_1\in \overline{\mathbb{R}}
\end{equation}
\tag{5.1}
$$
is well defined. If the function $\theta\underset{\theta\in [0,2\pi)}{\longmapsto} M(re^{i\theta})$ is Riemann integrable on $[0,2\pi)$ or can, for instance, be represented as the limit of a monotone sequence of continuous function, then it can be calculated as in (1.1). Theorem 5.1 (main theorem). Let $0<R_0\in \overline{\mathbb{R}}^+$, let $U$ be a difference of subharmonic functions on $D(R_0)$ distinct from $-\infty$ that is negative $\mathfrak{m}_2$-almost everywhere, and let some objects be selected as in (I) and (II) in Theorem 2.1 for $R<R_0$. Then the following inequality holds:
$$
\begin{equation}
\begin{aligned} \, \notag & \iint_{\overline D(R)\setminus \overline D(r)} \bigl(F(|z|)-F(R)\bigr)s(\arg z)\,\mathrm{d}\varDelta_U(z) \\ \notag &\qquad\qquad +\frac{F(r)-F(R)}{r^p}\iint_{\overline D(r)} |z|^ps(\arg z)\,\mathrm{d} \varDelta_U(z) \\ &\qquad \leqslant \|s\|_{\mathbb{R}} \bigl(p(F(r)-F(R))-rF'_{\mathrm{rh}}(r)\bigr)(-U)^{\circ r}. \end{aligned}
\end{equation}
\tag{5.2}
$$
Proof. Only the case $\|s\|_{\mathbb{R}}>0$ is of interest.
In view of (2.1) and since $F$ is decreasing on $[r,R]$, we have
$$
\begin{equation}
0\leqslant p\bigl(F(r)-F(R)\bigr)\leqslant pF(r)\overset{(2.1)}{<}+\infty\quad\text{and} \quad 0\leqslant -rF'_{\mathrm{rh}}(r)\overset{(2.1)}{<}+\infty.
\end{equation}
\tag{5.3}
$$
Since the difference $U$ of two subharmonic function distinct from $-\infty$ is negative $\mathfrak{m}_2$-almost everywhere, it is $\mathfrak{m}_1$-integrable on $\partial \overline D(r)$ and negative $\mathfrak{m}_1$-almost everywhere. In particular, $0\leqslant (-U)^{\circ r}<+\infty$, and all terms on the right-hand side of (5.2) are positive by (5.3).
We will use Corollary 3.1 in the case when
$$
\begin{equation}
L:=\partial \overline D(r), \qquad D_0:=D(r)\quad\text{and} \quad D_1:=D(R)\setminus \overline D(r)
\end{equation}
\tag{5.4}
$$
are the circle of radius $r$, the open disc of radius $r$ and the open annulus of radii $r<R$, with centre $0$, respectively, where $R<R_0$.
First we establish (5.2) for two twice continuously differentiable functions, $s$ on $\mathbb{R}$ and $F$ on $(r,R)$. Consider the positive function
$$
\begin{equation}
V_1\colon z\underset{0< |z|\leqslant R}{\longmapsto} s(\arg z)\bigl(F(|z|)-F(R)\bigr)
\end{equation}
\tag{5.5}
$$
constructed as in (4.31), By Proposition 4.13 $V_1$ belongs to the class $C^1(\overline D_1)\cap C^2( D_1)$, where $D_1\,{\overset{(5.4)}{=}}\,D(R)\setminus \overline D(r)$, and $V_1$ as $V$ has the properties (4.32) and (4.33). Thus, in the case (5.4) the function $V_1\geqslant 0$ satisfies conditions (i) and (iii) in Proposition 3.1, leaving aside condition (ii) for the moment. In the polar coordinates $z=te^{i\theta}$ the values of the derivative of $V_1$ on the circle $\partial \overline D(r)$ along the inward normal $\vec{\mathbf n}_{D_1} $ to $D_1\overset{(5.4)}{=}D(R)\setminus \overline D(r)$ are
$$
\begin{equation}
\frac{\partial V_1}{\partial \vec{\mathbf n}_{D_1} }(re^{i\theta})\overset{(5.5)}{=}\frac{\partial s(\theta)\bigl(F(t)-F(R)\bigr)}{\partial t}\biggm|_{t:=r} =s(\theta)F'(r).
\end{equation}
\tag{5.6}
$$
We also select a positive subharmonic function
$$
\begin{equation}
V_0\colon z\underset{ |z|\leqslant r}{\longmapsto} \frac{F(r)-F(R)}{r^p}s(\arg z)|z|^p
\end{equation}
\tag{5.7}
$$
in the class $C^1(\overline D_0)\cap C^2( D_0)$, where $D_0\overset{(5.4)}{=}D(r)$, such that $V_0=V_1$ on the circle $L\overset{(5.4)}{=}\partial \overline D(r)$ by construction. The values of the derivatives of $V_0$ on $\partial \overline D(r)$ along the inward normal $\vec{\mathbf n}_{D_0} $ to $D_0\overset{(5.4)}{=} D(r)$ are
$$
\begin{equation}
\frac{\partial V_0}{\partial \vec{\mathbf n}_{D_0} }(re^{i\theta})\overset{(5.7)}{=} -\frac{\partial (s(\theta)t^p(F(r)-F(R))/r^p)}{\partial t}\biggm|_{t:=r} = -p s(\theta)\frac{F(r)-F(R)}{r}.
\end{equation}
\tag{5.8}
$$
At $V_0$, in the polar coordinates $z=te^{i\theta}$ the Laplace operator (4.34) is
$$
\begin{equation*}
(\Delta V_0)(te^{i\theta})\overset{(5.7)}{=}\bigl(s''(\theta)+p^2s(\theta)\bigr) t^{p-2}\geqslant 0,
\end{equation*}
\notag
$$
because for the $p$-trigonometrically convex twice continuously differentiable function $s$ we have $s''+p^2s\geqslant 0$ (see [1], [10], Theorem 26, or [11], Theorem 7.15). Thus, all three assumptions (i)–(iii) of Proposition 3.1 are fulfilled. In combination with the positivity $V_0\geqslant 0$ on $D_0\overset{(5.4)}{=}D(r)$ and $V_1\geqslant 0$ on $\overline D(R)\setminus D(r)$, this allows us to apply Corollary 3.1 to the glued continuous function on $\mathbb{C}$
$$
\begin{equation*}
V(z)\overset{(3.1)}{=} s(\arg z)\cdot \begin{cases} \bigl(F(r)-F(R)\bigr)\dfrac{|z|^p}{r^p}&\text{for } |z|\leqslant r, \\ F(|z|)-F(R)&\text{for } r\leqslant |z|\leqslant R, \\ 0&\text{for } R\leqslant |z|< R_0. \end{cases}
\end{equation*}
\notag
$$
Hence for this function $V$, in view of (5.6)–(5.8) and since $F_R(R)=0$, inequality (3.5) can be detailed as
$$
\begin{equation}
\begin{aligned} \, \notag \iint_{D(R)}V\,\mathrm{d}\varDelta_U &=\iint_{\overline D(R)\setminus \overline D(r)}s(\arg z)\bigl(F(|z|)-F(R)\bigr) \,\mathrm{d}\varDelta_U(z) \\ \notag &\qquad +\frac{F(r)-F(R)}{r^p}\iint_{\overline D(r)}s(\arg z)|z|^p\,\mathrm{d}\varDelta_U(z) \\ \notag &\!\!\overset{(3.5)}{\leqslant} \frac{1}{2\pi}\int_L U\biggl(\frac{\partial V_0}{\partial \vec{\mathbf n}_{D_0} }+ \frac{\partial V_1}{\partial \vec{\mathbf n}_{D_1} }\biggr)\,\mathrm{d} \mathfrak{m}_1 \\ \notag &\!\!\!\!\!\!\!\!\overset{(5.6),\,(5.8)}{=} \frac{1}{2\pi}\int_0^{2\pi}U(re^{i\theta})\biggl(s(\theta)F'(r) -p s(\theta)\frac{F(r)-F(R)}{r}\biggr)r\,\mathrm{d} \theta \\ &=\bigl(p(F(r)-F(R))-rF'(r)\bigr) \frac{1}{2\pi}\int_0^{2\pi}\bigl(-U(re^{i\theta})\bigr)s(\theta)\,\mathrm{d} \theta. \end{aligned}
\end{equation}
\tag{5.9}
$$
Here the factor multiplying the last integral is positive in view of (5.3), and the factors $-U$ and $s$ in the integrand are positive by assumption, so the right-hand side of (5.9) is at most
$$
\begin{equation*}
\begin{aligned} \, &\bigl(p (F(r)-F(R))-rF'(r)\bigr)\|s\|_{\mathbb{R}} \frac{1}{2\pi}\int_0^{2\pi}\bigl(-U(re^{i\theta})\bigr)\,\mathrm{d} \theta \\ &\qquad \overset{(5.1)}{=}\|s\|_{\mathbb{R}}\bigl(p (F(r)-F(R))-rF'(r)\bigr) (-U)^{\circ r}. \end{aligned}
\end{equation*}
\notag
$$
In view of (5.9) this completes the proof of (5.2) for $s\in C^2(\mathbb{R})$ and $F\in C^2(r,R)$. Now we remove these constraints.
Let $F\in C^2(r,R)$ as before, but assume that $s$ is an arbitrary positive $p$-trigonometrically convex $2\pi$-periodic function on $\mathbb{R}$. Then there exists a sequence $(s_n)_{n\in \mathbb{N}}$ of twice continuously differentiable $p$-trigonometrically convex $2\pi$-periodic functions on $\mathbb{R}$ that converges to $s$ uniformly on $\mathbb{R}$ (see [15], Proposition 1.7, [10], Theorem 51, or [1], Lemma 16.4). Applying to such a pair $g$, $s_n$ inequality (5.2), verified already in this case, and letting $n\to +\infty$ yields inequality (5.2) without the condition $s\in C^2(\mathbb{R})$, but still for $F\in C^2(r,R)$. We can remove this constraint by using Proposition 4.11 for $p=0$ and Proposition 4.12 for $p>0$. In fact, let $(F_n)$ be a decreasing sequence of infinitely differentiable functions from Proposition 4.11 or 4.12 (depending on $p$) that converges to $F-F(R)$ uniformly on $[r,R]$, and such that $F_n(R)\underset{n\in \mathbb{N}}{=}0$ and $\lim_{n\to \infty}F_n'(r)\overset{(4.9)}{=}F'_{\mathrm{rh}}(r)$ at $r$. Then we have (5.2) for $F_n=F_n-F_n(R)$ as $F-F(R)$, and letting $n$ tend to infinity, from inequalities (5.2) for such $F_n$ we obtain (5.2) without extra assumptions about the differentiability properties of $s$ and $F$. Theorem 5.1 is proved. We can express inequality (5.2) in the main theorem in terms of Stieltjes integrals with respect to radial angular counting functions (1.13). In fact, if $f$ is an arbitrary bounded continuous function on $(r,R]\subset \mathbb{R}^+\setminus \{0\}$, then by the definition (1.13) and the definition of an integral we have the equality (see [18], Lemma 1)
$$
\begin{equation*}
\iint_{\overline D(R)\setminus \overline D(r)}f(|z|)s(\arg z)\,\mathrm{d} \varDelta(z) =\int_r^Rf(t)\,\mathrm{d} \varDelta^{\mathfrak{ra}(s)}(t).
\end{equation*}
\notag
$$
Applying it to the first integral on the left-hand side of (5.2) for $f:=F-F(R)$, and to the second integral there, taken over $(0,r]$, for $f\colon t\mapsto t^p$ we obtain the following result. Corollary 5.1. Under the assumptions of the main theorem (Theorem 5.1) 5.2. The subharmonic version and the proof of Theorem 2.1In (5.10), from Stieltjes integrals we can co over to Riemann ones. Theorem 5.2. Let $0<R_0\in \overline{\mathbb{R}}^+$, and let $p\in \mathbb{R}^+$ and $0<r<R<R_0$ be arbitrary numbers and the functions $s$ and $F$ be as in (I) and (II) of Theorem 2.1, and let $M$ and $u$ be subharmonic functions on $D(R_0)$ such that Then for $Q_{p,F}(r):=p(F(r)-F(R))-rF'_{\mathrm{rh}}(r)\overset{(2.1)}{<}+\infty$ the inequality
$$
\begin{equation}
\int_r^R (-F'_{\mathrm{rh}}(t)) \bigl(\varDelta_u^{\mathfrak{ra}(s)} (t)-\varDelta_M^{\mathfrak{ra}(s)} (t)\bigr)\,\mathrm{d} t \leqslant \|s\|_{\mathbb{R}} Q_{p,F}(r)\bigl(M^{\circ r}-u(0)\bigr)
\end{equation}
\tag{5.12}
$$
holds. If the function $F$ in (II) is positive, decreasing and $ p$-power convex on the whole of $(0, R)$, then, provided that the quantity ${\lim}_0^pF<+\infty$ in (2.4) is finite, inequality (5.12) holds for all $r\in (0,R)$ with coefficient s $r^{-p}\check{p}\,{\lim}_0^p F$ in place of $Q_{p,F}(r)$ on the right-hand side of (5.12), where $\check{p}$ is defined in (2.6). Proof. Set $U:=u-M$. Then all assumptions of the main theorem (Theorem 5.1) are fulfilled, and by Corollary 5.1 we have inequality (5.10). Integrating simultaneously by parts in the two integrals on the left-hand side of (5.10) and taking Proposition 4.4 into account, since $F-F(R)$ vanishes at $R$ and the radial-angular counting function is right continuous, in accordance with the convention (1.16), we have the equality
$$
\begin{equation*}
\begin{aligned} \, &\int_r^R\bigl(F(t)-F(R)\bigr)\,\mathrm{d} \varDelta_U^{\mathfrak{ra}(s)} (t)+\frac{F(r)-F(R)}{r^p}\int_0^r t^p\,\mathrm{d} \varDelta_U^{\mathfrak{ra}(s)} (t) \\ &\qquad =\int_r^R(-F'_{\mathrm{rh}}(t))\varDelta_U^{\mathfrak{ra}(s)} (t)\,\mathrm{d} t -\frac{F(r)-F(R)}{r^p}p\int_0^r t^{p-1}\varDelta_U^{\mathfrak{ra}(s)} (t) \,\mathrm{d} t. \end{aligned}
\end{equation*}
\notag
$$
Thus we can write (5.10) as the inequality
$$
\begin{equation}
\begin{aligned} \, \notag &\int_r^R(-F'_{\mathrm{rh}}(t))\varDelta_U^{\mathfrak{ra}(s)} (t)\,\mathrm{d} t \leqslant \frac{F(r)-F(R)}{r^p}p\int_0^r t^{p-1}\varDelta_U^{\mathfrak{ra}(s)} (t) \,\mathrm{d} t \\ &\qquad\qquad +\|s\|_{\mathbb{R}} \bigl(p( F(r)-F(R))-rF'_{\mathrm{rh}}(r)\bigr)(-U)^{\circ r}=:A+B, \end{aligned}
\end{equation}
\tag{5.13}
$$
where for $U=u-M$ the second term $B$ on the right-hand side has the form
$$
\begin{equation}
B=\|s\|_{\mathbb{R}} \bigl(p(F(r)-F(R))-rF'_{\mathrm{rh}}(r)\bigr)(M^{\circ r}-u^{\circ r}).
\end{equation}
\tag{5.14}
$$
For the first term $A$ on the right-hand side of (5.13), by the equality $\varDelta_U=\varDelta_u-\varDelta_M$ and the resulting inequality $\varDelta_U^{\mathfrak{ra}(s)}=\varDelta_u^{\mathfrak{ra}(s)}-\varDelta_M^{\mathfrak{ra}(s)}\leqslant \varDelta_u^{\mathfrak{ra}(s)}$ and by the first inequality in (5.3) we obtain
$$
\begin{equation*}
\begin{aligned} \, A &:= \frac{F(r)-F(R)}{r^p}p\int_0^r t^{p-1}\bigl(\varDelta_u^{\mathfrak{ra}(s)} (t)-\varDelta_M^{\mathfrak{ra}(s)} (t)\bigr) \,\mathrm{d} t \\ &\leqslant \frac{F(r)-F(R)}{r^p}p\int_0^r t^{p-1}\varDelta_u^{\mathfrak{ra}(s)} (t) \,\mathrm{d} t \leqslant p(F(r)-F(R))\int_0^r\frac{\varDelta_u^{\mathfrak{ra}(s)} (t)}{t}\,\mathrm{d} t. \end{aligned}
\end{equation*}
\notag
$$
Hence, as it obviously follows from the definitions (1.11) and (1.13) that
$$
\begin{equation*}
\varDelta_u^{\mathfrak{ra}(s)} (t)\overset{(1.13)}{\leqslant} \|s\|_{\mathbb{R}}\varDelta_u^{\mathfrak{ra}(1)} (t) = \|s\|_{\mathbb{R}}\varDelta_u^{\mathfrak{r}} (t),
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
A\leqslant \|s\|_{\mathbb{R}} p(F(r)-F(R))\int_0^r\frac{\varDelta_u^{\mathfrak{r}} (t)}{t}\,\mathrm{d} t.
\end{equation*}
\notag
$$
By the Poisson–Jensen–Nevanlinna–Privalov formula (1.2) we have
$$
\begin{equation*}
\int_0^r\frac{\varDelta_u^{\mathfrak{r}} (t)}{t}\,\mathrm{d} t\overset{(1.2)}{=}u^{\circ r}-u(0), \qquad u(0)\overset{(5.11)}{\neq} -\infty,
\end{equation*}
\notag
$$
which yields $A\leqslant \|s\|_{\mathbb{R}}p\bigl(F(r)-F(R)\bigr)\bigl(u^{\circ r}-u(0)\bigr)$. Hence from (5.14) we obtain
$$
\begin{equation*}
\begin{aligned} \, A+B &\leqslant \|s\|_{\mathbb{R}}p\bigl(F(r)-F(R)\bigr)(u^{\circ r}-u(0)) \\ &\qquad +\|s\|_{\mathbb{R}} \bigl(p(F(r)-F(R))-rF'_{\mathrm{rh}}(r)\bigr)(M^{\circ r}-u^{\circ r}) \\ &= \|s\|_{\mathbb{R}} \bigl(p(F(r)-F(R))-rF'_{\mathrm{rh}}(r)\bigr) (M^{\circ r}-u(0)) +\|s\|_{\mathbb{R}}rF'_{\mathrm{rh}}(r)(u^{\circ r}\,{-}\,u(0)). \end{aligned}
\end{equation*}
\notag
$$
Here the second term is negative because, since $F$ is decreasing, the factor $F'_{\mathrm{rh}}(r)\overset{(5.3)}{\leqslant} 0$ is negative, and the integral mean value on the circle $u_r^{\circ}$ is no less than $u(0)$ in the case when $u$ is subharmonic. Thus, the sum $A+B$ in (5.13) does not exceed the right-hand side of (5.12). Substituting $\varDelta_u^{\mathfrak{ra}(s)} (t)-\varDelta_M^{\mathfrak{ra}(s)} (t)=\varDelta_U^{\mathfrak{ra}(s)} (t)$ into the left-hand integrand in (5.13) we obtain (5.12).
In the case of a decreasing positive $p$-power convex (for $p>0$) or $\log$-convex (for $p=0$) function $F$ on the whole interval $(0, R_0)$, in view of (5.12) it suffices to bound $Q_{p,F}(r)$ above by $r^{-p}\check{p}\,{\lim}_0^p F$, which we did already in inequality (4.17) in Proposition 4.8 for $p=0$ and in inequality (4.20) in Proposition 4.9 for $p>0$. Theorem 5.2 is proved. Proof of Theorem 2.1. Here we have $R_0:=+\infty$ and $D(R_0):=\mathbb{C}$. Set $u:=\log|f|$, so that $u\leqslant M$ on $\mathbb{C}$ by the hypotheses of Theorem 2.1. The Riesz mass distribution $\varDelta_u=\varDelta_{\log|f|}$ is defined in terms of the distribution of zeros $\operatorname{Zero}_f$ by
$$
\begin{equation*}
\varDelta_u(S) =\varDelta_{\log|f|}(S)=\sum_{z\in S}\operatorname{Zero}_f(z) \quad\text{for each } S\subset \mathbb{C}
\end{equation*}
\notag
$$
(see [27], Theorem 3.7.8), and for radial-angular counting functions, in view of $\operatorname{Zero}_f\geqslant Z$ we have
$$
\begin{equation}
\varDelta_u^{\mathfrak{ra}(s)}=\varDelta_{\log|f|}^{\mathfrak{ra}(s)} \overset{(1.8)}{=}\operatorname{Zero}_f^{\mathfrak{ra}(s)}\overset{(1.6)}{\geqslant} Z^{\mathfrak{ra}(s)}.
\end{equation}
\tag{5.15}
$$
Under the hypotheses of Theorem 2.1 the assumptions of Theorem 5.2 are fulfilled in this notation, and so its conclusions hold, where $\varDelta_u^{\mathfrak{ra}(s)}$ can by (5.15) be replaced by $\operatorname{Zero}_f^{\mathfrak{ra}(s)} \geqslant Z^{\mathfrak{ra}(s)}$. However, the integrand $-F'_{\mathrm{rh}}$ is positive because $F$ is decreasing, so we can replace $\varDelta_u^{\mathfrak{ra}(s)}$ by $Z^{\mathfrak{ra}(s)}\overset{(5.15)}{\leqslant} \varDelta_u^{\mathfrak{ra}(s)}$ in these conclusions.
Theorem 2.1 is proved. 5.3. Uniqueness distributions and the proof of Theorem 2.3Let $M$: ${D\to \overline{\mathbb{R}}}$ be some extended real function on a domain $D\subset\mathbb{C}$. We call a distribution of points $Z$ on $D$:
Here is a summary of elementary relations between these distributions. Proposition 5.1. Let $M\colon D\to \overline{\mathbb{R}}$ be a function on a domain $D\subset \mathbb{C}$. Then the following hold. 1. A distribution of zeros with respect to $M$ that has a finite multiplicity at some point in $D$ is a subdistribution of zeros with respect to $M$. The converse implication fails. 2. The notions of a subdistribution of zeros and a nonuniqueness distribution with respect to $M$ are equivalent. 3. A subdistribution of zeros with respect to $M$ is also a subdistribution of zeros with respect to $M+\log|h|$ for each nontrivial holomorphic function $h\neq 0$ on $D$. 4. A distribution of zeros with respect to $M$ is also a distribution of zeros with respect to $M+\log|h|$ for each holomorphic function $h$ not vanishing on $D$. 5. Let $H$ be a harmonic function on a simply or multiply connected domain $D\subset \mathbb{C}$ that has exterior points, that is, the complement $\mathbb{C}\setminus \overline D$ has a nonempty interior. Then a distribution of points $Z$ on $D$ is a distribution of zeros or a uniqueness distribution with respect to $M$ if and only of $Z$ is a distribution of zeros or a uniqueness distribution with respect to $M +H$, respectively. Proof. 1. If $f$ is a holomorphic function on $D$ satisfying the constraints (5.16) and $\operatorname{Zero}_f=Z$, and if $Z(z)<+\infty$ at some point $z\in D$, then $f$ is a nontrivial function and $Z$ is obviously a subdistribution of zeros with respect to $M$. This implication is irreversible as we see from the simple example when
$$
\begin{equation}
D:=\mathbb{C}, \qquad M\colon z\underset{z\in \mathbb{C}}{\longmapsto}|z|\quad\text{and} \quad Z\colon z\underset{z\in \mathbb{C}}{\longmapsto} \begin{cases} 1&\text{for } z\in \mathbb{N}, \\ 0&\text{for } z\notin \mathbb{N}, \end{cases}
\end{equation}
\tag{5.18}
$$
in combination with Lindelöf’s theorem on the description of the distributions of zeros of entire functions of finite type with respect to an integer order (see [1], Ch. I, § 11, Theorem 15).
2. If $Z$ is a subdistribution of zeros with respect to $M$, then a nontrivial holomorphic function vanishing on $Z$ and satisfying (5.16), in combination with the trivial function $g=0$, shows that $Z$ is a nonuniqueness distribution with respect to $M$. Conversely, if $Z$ is a uniqueness distribution with respect to $M$, then there exists a pair of holomorphic functions $f\neq g$ on $D$ that satisfies (5.17). Then $ F:=\frac12(f-g)\neq 0$ vanishes on $Z$ and
$$
\begin{equation*}
\log |F|\leqslant \log\bigl(|f|+|g|\bigr)-\log 2\overset{(5.17)}{\leqslant}M.
\end{equation*}
\notag
$$
Hence $Z$ is a subdistribution of zeros with respect to $M$.
3–4. Let $f$ be a realization of the (sub)distribution of zeros $Z$ with respect to $M$. Then the product $fh$ is a realization of the (sub)distribution of zeros $Z$ with respect to $M+\log|h|$. 5. By part 2, we will have proved assertion 5 once we will have shown that the (sub)distributions of zeros with respect to $M$ and with respect to $M \pm H$ are the same. Let $f$ be a realization of a (sub)distributions of zeros with respect to $M$. As shown in Lemma 2.1 in [39], in domains $D$ of the above form, for each harmonic function $H$ there exists a holomorphic function $h$ with no zeros in $D$ (that is, with $\operatorname{Zero}_h=0$) such that $\log |h|\leqslant H$ on $D$. Hence the holomorphic product $fh$ is a realization in $D$ of a (sub)distribution of zeros with respect to $M+H$, and the holomorphic ratio $f/h$ is a realization of a (sub)distribution of zeros with respect to $M-H$. Proposition 5.1 is proved. Remark 5.1. If $M\neq -\infty$ is a subharmonic or a $\delta$-subharmonic function, then part 5 of Proposition 5.1 shows that (non)uniqueness distributions (that is, distributions of zeros) with respect to $M$ do not depend on $M$ itself, but only of the Riesz charge or mass distribution $\varDelta_M$. Proof of Theorem 2.3. We prove an equivalent result, the converse of the opposite of Theorem 2.3. By part 2 of Proposition 5.1, to do this we must show that, given a subdistribution of zeros $Z$ with respect to a function $M$ satisfying (2.32) for $\varDelta_M^{\mathfrak{r}\circ}$ in place of $\varDelta^{\mathfrak{r}\circ}$ and satisfying the inequality $\varDelta_M^{\mathfrak{ra}(s)}\leqslant \varDelta^{\mathfrak{ra}(s)}$ for all $s\in T_p$, the supremum in (2.31) is finite.
Let $f_Z\neq 0$ be an entire function realizing the subdistribution of zeros $Z$ with respect to $M$. If $f_Z(0)\neq 0$, that is, $\operatorname{Zero}_{f_Z}(0)=0$, then set $f:=f_Z$. If $f_Z(0)=0$ and $n:=\operatorname{Zero}_f(0)$ is the multiplicity of the zero of $f_Z$ at the origin, then we consider the function $f\colon z\longmapsto af_{Z}(z)/z^n$, where $0<a\in \mathbb{R}^+$. By construction $f(0)\neq 0$, $f$ vanishes on $Z$, and as $M$ is bounded in a neighbourhood of the origin, by the maximum principle for $\log|f|$ the positive coefficient $a$ can be taken sufficiently small so that $\log|f|\leqslant M$ on $\mathbb{C}$. Thus, in any case the entire function $f$ is a realization of $Z$ with respect to $M$. Then by the second part of Theorem 2.1, for each choice of functions $s$ and $F$ as in items (I) and (II) in Theorem 2.1 we have (2.5). In particular, for each $s\in T_p\subset p\text{-}\mathrm{trc}^+$, in combination with an arbitrary function $F\in p\text{-}\mathrm{pwc}^{+\downarrow}_1(0,R)$, in view of the normalization ${\lim}_0^pF=1$, for each $r\in (0,R)$ we have
$$
\begin{equation*}
\int_r^R \bigl(Z^{\mathfrak{ra}(s)}(t)-\varDelta_M^{\mathfrak{ra}(s)}(t)\bigr) (-F'_{\mathrm{rh}}(t))\,\mathrm{d} t \leqslant \check{p}\|s\|_{\mathbb{R}}\frac{M^{\circ r}-\log|f(0)|}{r^p}.
\end{equation*}
\notag
$$
By constraint (2.32), where the function $m>0$ is bounded away from zero, there exists a constant $C\in \mathbb{R}^+$ independent of $R$ such that $M^{\circ r}-\log|f(0)|\leqslant Cm(r)$ for all $r >0$. Hence, as $\varDelta_M^{\mathfrak{ra}(s)}\leqslant \varDelta^{\mathfrak{ra}(s)}$ for all $s\in T_p$ and since $ (-F'_{\mathrm{rh}}(t))\geqslant 0$ because $F$ is decreasing, we obtain
$$
\begin{equation*}
\int_r^R \bigl(Z^{\mathfrak{ra}(s)}(t)- \varDelta^{\mathfrak{ra}(s)}(t)\bigr) (-F'_{\mathrm{rh}}(t))\,\mathrm{d} t \leqslant \check{p}\|s\|_{\mathbb{R}}\frac{Cm(r)}{r^p} \quad \text{for all } r\in (0,R).
\end{equation*}
\notag
$$
Division of both sides by $\check{p}\|s\|_{\mathbb{R}}m(r)/r^p$ for $s\neq 0$ yields
$$
\begin{equation*}
\frac{1}{m(r)}r^p\int_r^R \bigl(Z^{\mathfrak{ra}(s)}(t)- \varDelta^{\mathfrak{ra}(s)}(t)\bigr) \frac{-F'_{\mathrm{rh}}(t)}{\check{p}\|s\|_{\mathbb{R}}} \,\mathrm{d} t \leqslant C \quad \text{for all } r\in (0,R),
\end{equation*}
\notag
$$
where $C\in \mathbb{R}^+$ is independent of the choice of $s$, $r, R$ and $F$. Thus we have shown that the double supremum in (2.31) is finite under the assumptions made in the beginning of the proof. This completes the proof of Theorem 2.3. AcknowledgementI am deeply obliged to the referees, whose thorough reviews helped me to fix a number of inaccuracies and, hopefully, improve the presentation significantly.
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