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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
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| Abstract |
In the paper we consider entire functions
f:Cp→C, p≥2,
defined by power series
f(z)=f(z1,…,zp)=∑‖
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.
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| Keywords |
entire functions of several variables; multiple power series; maximum modulus; maximal term;
Wiman’s inequality |
| Reference |
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(in Russian)
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| Pages |
35-50 |
| Volume |
38 |
| Issue |
1 |
| Year |
2012 |
| Journal |
Matematychni Studii |
| Full text of paper |
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