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Wiman-Type Inequality in Multiple-Circular Domains: Lévy’s Phenomenon and Exceptional Sets

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For the classical Wiman inequality Mf(r) ≤ μf(r)(lnμf(r))1/2 + ε, ε > 0, with entire functions \( f(z)={\sum}_{n=0}^{+\infty }{a}_{n}{z}^{n},z\in \mathbb{C}, \) which is true outside a set of finite logarithmic measure, P. Lévy established (1929) that, under certain additional regularity conditions imposed on ln Mf (r), the constant 1/2 can be replaced with 1/4 almost surely in a certain probability sense; here, Mf(r) = max {|f(z)| : |z| = r}, μf(r) = max {|an|rn :  n ≥ 0}, r > 0. We prove that the result established by Lévy remains true in the case of Wiman-type inequality for analytic functions in any multiply circular domain. This gives an affirmative answer to the question posed by A. Goldberg and M. Sheremeta in (1996). Earlier, affirmative answers to this question were obtained in the case of Fenton’s inequality for the entire functions of two variables (Mat. Stud., 23, No. 2 (2005)), for the entire functions of several variables (Ufa Math. J., 6, No. 2 (2014)), and for the analytic functions of several variables in a polydisc (Eur. J. Math., 6, No. 1 (2020)).

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  1. I. Franko National University of Lviv, Lviv, Ukraine

    A. O. Kuryliak & O. B. Skaskiv

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  1. A. O. Kuryliak
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Correspondence to A. O. Kuryliak.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 5, pp. 650–661, May, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i5.7137.

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Kuryliak, A.O., Skaskiv, O.B. Wiman-Type Inequality in Multiple-Circular Domains: Lévy’s Phenomenon and Exceptional Sets. Ukr Math J 74, 743–756 (2022). https://doi.org/10.1007/s11253-022-02098-y

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  • DOI: https://doi.org/10.1007/s11253-022-02098-y

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