The convergence classes for analytic functions in the Reinhardt domains

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 $$\sum_{k=0}^{+\infty}(\alpha(k)-\alpha(k-1)) \beta_1\left({k}/{\ln^{+}|A_k|}\right)<+\infty,$$ $$\beta_1(x)= \int\limits_{x}^{+\infty} \frac{dt}{\beta(t)},\quad A_k=\max\{|a_n|\colon\|n\|=k\}. $$

Keywords

• analytic function
• reinhardt domain
• convergence class