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Ufimsk. Mat. Zh., 2021, Volume 13, Issue 1, Pages 69–77 (Mi ufa553)  

Extremal problems in theory of central Wiman-Valiron index

K. G. Malyutinab, M. V. Kabankoa, V. A. Malyutincd

a Kursk State University, Radischeva str. 33, 305000, Kursk, Russia
b South-West State University, 50 let Oktyabrya str. 94, 305040, Kursk, Russia
c Sumy State University, Rimskogo-Korsakova str. 2, 40007, Sumy, Ukraine
d Riverstone International School, 5521 East Warm Springs Avenue, Boise, ID 83716 United States of America

Abstract: We consider some properties of central index in Wiman-Valiron index. We introduce the notion of a determining sequence of a central index $\nu(r)$ corresponding to a fixed transcendental function $f$ and the notion of a determining sequence for an arbitrary fixed central index $\nu(r)$. Let $\rho_1,\rho_2,…,\rho_s,…$ be the points of the jumps of the function $\nu(r)$ taken counting their multiplicities. This means that if at a point $\rho_s$ the jump is equal to $m_s$, then the quantity $\rho_s$ appears $m_s$ times in this sequence. Such sequence is called determining sequence of the function $\nu(r)$. We introduce the notion of the regularization of the function $\nu(r)$, which is employed for proving main statements. We study two extremal problems in the class of functions with a prescribed central index. We obtain the expression for the maximum of the modulus of the extremal function in terms of its central index. The main obtained results are as follows. Let $T_\nu$ be the set of all transcendental functions $f$ with a prescribed central index $\nu(r)$, $M(r,f)=\max\{|f(re^{i\theta})|:  0\leqslant\theta\leqslant2\pi\}$, and let $M(r,\nu)=\sup\{M(r,f):f\in T_\nu\}$. Then for each $r>0$, in the class of the functions $T_{\nu}$, the quantity $M(r,\nu)$ is attained at the same function for all $r>0$. We describe the form of such extremal function. We also prove that for each fixed $r_0>0$ and for each prescribed central index $\nu(r)$, in the class $T_\nu$ there exists a function $f_0(z)$ such that $M(r_0,f_0)=\inf\{M(r_0,f):f\in T_\nu\}$.

Keywords: Wiman-Valiron theory, central index, determining sequence, regularization, extremal problem.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00236
The reported study by was funded by RFBR according to the research project no. 18-01-00236.

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English version:
Ufa Mathematical Journal, 2021, 13:1, 68–76 (PDF, 333 kB); https://doi.org/10.13108/2021-13-1-68

Bibliographic databases:

UDC: 517.547
MSC: 30D10, 30D20
Received: 03.12.2020

Citation: K. G. Malyutin, M. V. Kabanko, V. A. Malyutin, “Extremal problems in theory of central Wiman-Valiron index”, Ufimsk. Mat. Zh., 13:1 (2021), 69–77; Ufa Math. J., 13:1 (2021), 68–76

Citation in format AMSBIB
\by K.~G.~Malyutin, M.~V.~Kabanko, V.~A.~Malyutin
\paper Extremal problems in theory of central Wiman-Valiron index
\jour Ufimsk. Mat. Zh.
\yr 2021
\vol 13
\issue 1
\pages 69--77
\jour Ufa Math. J.
\yr 2021
\vol 13
\issue 1
\pages 68--76

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