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 Ufimsk. Mat. Zh., 2014, Volume 6, Issue 2, Pages 113–122 (Mi ufa245)

Levy's phenomenon for entire functions of several variables

A. O. Kuryliak, O. B. Skaskiv, O. V. Zrum

Department of Mechanics and Mathematics, Ivan Franko National University of L'viv, Universytets'ka str. 1, 79000, Lviv, Ukraine

Abstract: For entire functions $f(z)=\sum_{n=0}^{+\infty}a_nz^n$, $z\in\mathbb C$, P. Lévy (1929) established that in the classical Wiman's inequality $M_f(r)\leq\mu_f(r)(\ln\mu_f(r))^{1/2+\varepsilon}$, $\varepsilon>0$, 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 $M_f(r)=\max\{|f(z)|\colon|z|=r\}$, $\mu_f(r)=\max\{|a_n|r^n\colon n\geq0\}$, $r>0$. In this paper we prove that the phenomenon discovered by P. Lévy 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.

Keywords: Levy's phenomenon, random entire functions of several variables, Wiman's inequality.

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English version:
Ufa Mathematical Journal, 2014, 6:2, 111–120 (PDF, 392 kB); https://doi.org/10.13108/2014-6-2-111

UDC: 517.55
MSC: 30B20, 30D20
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Citation: A. O. Kuryliak, O. B. Skaskiv, O. V. Zrum, “Levy's phenomenon for entire functions of several variables”, Ufimsk. Mat. Zh., 6:2 (2014), 113–122; Ufa Math. J., 6:2 (2014), 111–120

Citation in format AMSBIB
\Bibitem{KurSkaZru14} \by A.~O.~Kuryliak, O.~B.~Skaskiv, O.~V.~Zrum \paper Levy's phenomenon for entire functions of several variables \jour Ufimsk. Mat. Zh. \yr 2014 \vol 6 \issue 2 \pages 113--122 \mathnet{http://mi.mathnet.ru/ufa245} \elib{http://elibrary.ru/item.asp?id=21596979} \transl \jour Ufa Math. J. \yr 2014 \vol 6 \issue 2 \pages 111--120 \crossref{https://doi.org/10.13108/2014-6-2-111} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84928200847} 

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This publication is cited in the following articles:
1. A. O. Kuryliak, V. L. Tsvigun, “Wiman's inequality for analytic functions in $\mathbb D\times\mathbb C$ with rapidly oscillating coefficients”, Carpathian Math. Publ., 10:1 (2018), 133–142
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