Article
Ufa Mathematical Journal
Volume 6, Number 2, pp. 111-120
Levy's phenomenon for entire functions of several variables
Kuryliak A.O. , Skaskiv O.B., Zrum O.V.
DOI:10.13108/2014-6-2-111
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For entire functions $f(z)=\sum_{n=0}^{+\infty}a_nz^n,\ z\in \mathds{C},$ P. L${\rm \acute{e}}$vy (1929) established that in the classical Wiman's inequality $M_f(r)\leqslant\mu_f(r)(\ln\mu_f(r))^{1/2+\varepsilon},\ \varepsilon>0,$ which holds outside a set of finite logarithmic measure, the constant $1/2$ can be replaced almost surely in some sense by $1/4$; here $M_f(r)=\max\{|f(z)|\colon |z|=r\},\ \mu_f(r)=\max\{|a_n|r^n\colon n\geqslant0\},\ r>0. $ In this paper we prove that the phenomenon discovered by P. L${\rm\acute{e}}$vy holds also in the case of Wiman's inequality for entire functions of several variables, which gives an affirmative answer to the question of A. A. Goldberg and M. M. Sheremeta (1996) on the possibility of this phenomenon.