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ЖУРНАЛЫ // Проблемы анализа — Issues of Analysis // Архив

Пробл. анал. Issues Anal., 2022, том 11(29), выпуск 2, страницы 59–71 (Mi pa352)

Analytic functions of infinite order in half-plane

K. G. Malyutina, M. V. Kabankoa, T. V. Shevtsovab

a Kursk State University, 33 Radischeva str., Kursk 305000, Russia
b Southwest State University, 50 Let Oktyabrya Street, 94, Kursk 305040, Russia

Аннотация: J. B. Meles (1979) considered entire functions with zeros restricted to a finite number of rays. In particular, it was proved that if $f$ is an entire function of infinite order with zeros restricted to a finite number of rays, then its lower order equals infinity. In this paper, we prove a similar result for a class of functions analytic in the upper half-plane. The analytic function $f$ in $\mathbb{C}_+=\{z:\Im z>0\}$ is called proper analytic if $\limsup\limits_{z\to t}\ln|f(z)|\leq 0$ for all real numbers $t\in\mathbb{R}$. The class of the proper analytic functions is denoted by $JA$. The full measure of a function $f\in JA$ is a positive measure, which justifies the term "proper analytic function". In this paper, we prove that if a function $f$ is the proper analytic function in the half-plane $\mathbb{C}_+$ of infinite order with zeros restricted to a finite number of rays $\mathbb{L}_k$ through the origin, then its lower order equals infinity.

Ключевые слова: half-plane, proper analytic function, infinite order, lower order, Fourier coefficients, full measure.

УДК: 517.537

MSC: 30D35

Поступила в редакцию: 10.11.2021
Исправленный вариант: 03.05.2022
Принята в печать: 04.05.2022

Язык публикации: английский

DOI: 10.15393/j3.art.2022.11010



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