Wiman-type inequality in multiple-circular domains: Lévy’s phenomenon and exceptional sets
Abstract
УДК 517.555
For the classical Wiman inequality $M_f(r)\leq\mu_f(r)(\ln\mu_f(r))^{1/2+\varepsilon},$ $\varepsilon>0,$ with entire functions $f(z)=\displaystyle \sum\nolimits _{n=0}^{+\infty}a_nz^n,$ $z\in {\mathbb C},$ which holds outside a set of finite logarithmic measure, P. L${\rm\acute{e}}$vy established (1929) that under some additional regularity conditions on $\ln M_f (r)$ the constant $1/2$ can be replaced by $1/4$ almost surely in some sense; here $M_f(r)=\max \big \{|f(z)|\colon |z|=r \big \},$ $\mu_f(r)=\max \big \{|a_n|r^n\colon n\geq 04\},$ $r>0. $ In this paper, we prove that the result established by P. L${\rm\acute{e}}$vy holds also in the case of Wiman-type inequality for analytic functions in any multiple-circular domain, which gives an affirmative answer to the question posed by A. A. Goldberg and M. M. Sheremeta (1996). Earlier, the answer to their question was obtained for Fenton's inequality in the case of entire functions of two variables (Mat. Stud., {\bf 23}, \No 2 (2005)), entire functions of several variables (Ufa Math. J., {\bf 6}, \No 2 (2014)), and analytic functions of several variables in a polydisc (Eur. J. Math., {\bf 6}, \No 1 (2020)).
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