Wiman’s type inequalities without exceptional sets for random entire functions of several variables

Author A. O. Kuryliak, O. B. Skaskiv
matstud@franko.lviv.ua, kurylyak88@gmail.com
Ivan Franko National University of L’viv

Abstract In the paper we consider entire functions $f\colon \mathbb{C}^p\to\mathbb{C},\ p\geq 2,$ defined by power series $f(z)=f(z_1,\ldots,z_p)=\sum_{\|n\|=0}^{+\infty}a_n z^n, %\ p\geq2, z^n=z_1^{n_1}\cdot\ldots\cdot z_p^{n_p},$ $n=(n_1,\ldots,n_p).$ For $r=(r_1,\ldots ,r_p)\in\mathbb{R}_+^p$ we set $M_f(r)=\max\{|f(z)|\colon |z_i|\leq r_i, i\in\{1,\ldots,p\}\}, \ \mu_f(r)=\max\{|a_n|r^{n}\colon n\in\mathbb{Z}_+^p\},$ $ r^{\vee}=\max\{r_i\colon i\in\{1,\ldots,p\}\},\ r^{\wedge}=\min\{r_i\colon i\in\{1,\ldots,p\}\}$ and let $l$ be a log-convex real function on $(1,+\infty)$ such that $\ln t=o(l(t)),\ t\to+\infty.$ Then for any entire transcendental function $f$ {with} $\ln M_f(r)\leq l(r^{\vee}),\ r^{\wedge}\to+\infty,$ {the} inequality $\varlimsup\limits_{r^{\wedge}\to+\infty} \frac{\ln M_f(r)-\ln\mu_f(r)}{\ln\ln\mu_f(r)}\leq\alpha$ holds if and only if $ \varlimsup\limits_{t\to+\infty}(\ln l(t)/\ln\ln t)\leq1+\alpha/p.$ Similar theorems are proved for random entire functions of several complex variables.
Keywords entire functions of several variables; multiple power series; maximum modulus; maximal term; Wiman’s inequality
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Pages 35-50
Volume 38
Issue 1
Year 2012
Journal Matematychni Studii
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