# Wiman’s inequality for entire functions of several complex variables with rapidly oscillating coefficients

Author
Ivan Franko National University of Lviv
Abstract
Let $\mathcal{E}^p$ be a class of entire functions of the form $f(z)=\sum_{\|n\|=0}^{+\infty}a_nz^n$, $\|n\|=n_1+\ldots+n_p$ ($p\geq 2$), $z=(z_1,\ldots,z_p)\in\mathbb{C}^p$, and $\mathcal{K}(f,\theta)=\{f(z,t)=\sum_{\|n\|=0}^{+\infty}a_n \exp\{2\pi i\theta_n t\}r^n\colon t\in\mathbb R\}$, where $\{\theta_n\}$ is a sequence of positive {integers} such that its arrangement $\{\theta_k^*\}$ by increasing, i.e. $\{\theta_n\colon n\in \mathbb{Z}_+^p\}=\{\theta_k^*\colon k\geq 0\}$, $\theta_{k+1}^*>\theta_k^*$, satisfies the condition $\theta_{k+1}^*/\theta_k^*\geq q>1$ $(k\geq0)$. In this paper it is established that for $f\in \mathcal{E}^p$ almost surely for $t\in\mathbb R$ there {exists} a set $E(t)\subset\mathbb{R}_+^p$, such that for all $r\in\mathbb{R}_+^p \setminus E(t)$ the inequality $$\mathfrak{M}_f(r,t)=\max \limits_{|z|\leq r}|f(z,t)|\leq \mu_f(r)(\Lambda_f(r))^{1/4}\ln^{3} \Lambda_f(r)$$ holds, where $E(t)$ is a set of finite asymptotically logarithmic measure and $M_f(r)=\max\{|f(z,t)|\colon|z_i|=r_i,\ i\in \{1,\ldots,p\}\},$\ $\mu_f(r)=\max_{n\in\mathbb{Z}_+^p}\{|a_n|r^n\colon r=(r_1,\ldots,r_p)\in\mathbb{R}_+^p\},$ $\Lambda_f(r)=\ln^p\mu_f(r)\cdot\prod_{i=1}^p\ln^{p-1}r_i.$
Keywords
Wiman-Valiron’s inequality; random entire function of several variables
DOI
doi:10.15330/ms.43.1.16-26
Reference
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Pages
16-26
Volume
43
Issue
1
Year
2015
Journal
Matematychni Studii
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