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 Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 4, Pages 833–850 (Mi izv1276)

On Wiman's theorem concerning the minimum modulus of a function analytic in the unit disk

Abstract: This paper contains an investigation of conditions under which an analytic function $F(z)$ represented by a Dirichlet series
$$F(z)=\sum_{n=0}^\infty a_ne^{z\lambda_n},\qquad 0=\lambda_0<\lambda_n\uparrow+\infty\quad(n\to+\infty),$$
absolutely convergent in $ż\colon\operatorname{Re}z<0\}$ satisfies the relation
$$F(x+iy)=(1+o(1))a_{\nu(x)}e^{(x+iy)\lambda_{\nu(x)}}$$
uniformly with respect to $y\in\mathbf R$ as $x\to-0$ in the complement of some sufficiently small set. The results are used to derive as simple corollaries new assertions for functions analytic in the unit disk that are represented by lacunary power series. All the assertions proved in this article are best possible or close to best possible.
Bibliography: 12 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1990, 35:1, 165–182

Bibliographic databases:

UDC: 517.535
MSC: Primary 30B50; Secondary 30B10

Citation: O. B. Skaskiv, “On Wiman's theorem concerning the minimum modulus of a function analytic in the unit disk”, Izv. Akad. Nauk SSSR Ser. Mat., 53:4 (1989), 833–850; Math. USSR-Izv., 35:1 (1990), 165–182

Citation in format AMSBIB
\Bibitem{Ska89} \by O.~B.~Skaskiv \paper On Wiman's theorem concerning the minimum modulus of a~function analytic in the unit disk \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1989 \vol 53 \issue 4 \pages 833--850 \mathnet{http://mi.mathnet.ru/izv1276} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1018750} \zmath{https://zbmath.org/?q=an:0698.30025} \transl \jour Math. USSR-Izv. \yr 1990 \vol 35 \issue 1 \pages 165--182 \crossref{https://doi.org/10.1070/IM1990v035n01ABEH000694} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. B. Skaskiv, “On the minimum of the absolute value of the sum for a Dirichlet series with bounded sequence of exponents”, Math. Notes, 56:5 (1994), 1177–1184
2. A. M. Gaisin, “Behavior of the logarithm of the modulus value of the sum of a Dirichlet series converging in a half-plane”, Russian Acad. Sci. Izv. Math., 45:1 (1995), 175–186
3. A. M. Gaisin, “Maximal Term of the Modified Dirichlet Series”, Math. Notes, 69:6 (2001), 756–769
4. A. M. Gaisin, T. I. Belous, “Estimation over curves of the functions given by Dirichlet series on a half-plane”, Siberian Math. J., 44:1 (2003), 22–36
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