The convergence classes for analytic functions in the Reinhardt domains

  • T.M. Salo Lviv Polytechnic National University, 12 Bandera str., 79013, Lviv, Ukraine
  • O.Yu. Tarnovecka National Technical University “Kharkiv Polytechnic Institute”, 2, Kyrpychovastr. 61002, Kharkiv, Ukraine
Keywords: analytic function, Reinhardt domain, convergence class
Published online: 2018-12-31

Abstract


Let $L^0$ be the class of positive increasing on $[1,+\infty)$ functions $l$ such that $l((1+o(1))x)=(1+o(1))l(x)$ $(x\to +\infty)$. We assume that $\alpha$ is a concave function such that $\alpha(e^x)\in L^0$ and function $\beta\in L^0$ such that $\displaystyle\int_1^{+\infty}\frac{\alpha(x)}{\beta(x)}dx<+\infty$. In the article it is proved the following theorem: if $\displaystyle f(z)=\sum_{\|n\|=0}^{+\infty}a_nz_n$, $z\in \mathbb{C}^p$, is analytic function in the bounded Reinhard domain $G\subset \mathbb{C}^p$, then the condition $\displaystyle \int\limits_{R_0}^{1} \frac{\alpha(\ln^{+} M_{G}(R,f))} {(1-R)^2\beta(1/(1-R))}d\,R<+\infty,$ $M_{G}(R,f)=\sup\{|F(Rz)|\colon z\in G\},$ yields that $$\sum_{k=0}^{+\infty}(\alpha(k)-\alpha(k-1)) \beta_1\left({k}/{\ln^{+}|A_k|}\right)<+\infty,$$ $$\beta_1(x)= \int\limits_{x}^{+\infty} \frac{dt}{\beta(t)},\quad A_k=\max\{|a_n|\colon\|n\|=k\}. $$

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How to Cite
(1)
Salo T., Tarnovecka O. The Convergence Classes for Analytic Functions in the Reinhardt Domains. Carpathian Math. Publ. 2018, 10 (2), 408-411.