Abstract
Let \(\mathbb{C}\) be the complex plane, E be a measurable subset of a segment \([0, R]\) of the positive semiaxis \(\mathbb{R}^+\), and \(u\not\equiv - \infty\) be a subharmonic function on \(\mathbb{C}\). The main result of this article is an upper estimate of the integral of the module \(|u|\) over a subset of E through the maximum of the function u on a circle of radius R centered at zero and the linear Lebesgue measure of subset E. Our result develops one of the classical theorems of R. Nevanlinna, established in the case of \(E=[0, R]\), and versions of so-called small arcs lemma by Edrei–Fuchs for small intervals on \(\mathbb{R}^+\) from the works of A.F. Grishin, M.L. Sodin, and T.I. Malyutina. Our estimate is uniform in the sense that the constants in the inequality are absolute and do not depend on the subharmonic function u, under the semi-normalization \(u(0)\geq 0\).
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ACKNOWLEDGMENTS
The authors would like to express their sincere gratitude to the anonymous referee for helpful remarks.
Funding
The research is fulfilled under the financial support of the Russian Science Foundation (project no. 18-11-00002).
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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 9, pp. 15–24.
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Gabdrakhmanova, L.A., Khabibullin, B.N. A Small Intervals Theorem for Subharmonic Functions. Russ Math. 64, 12–20 (2020). https://doi.org/10.3103/S1066369X20090029
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DOI: https://doi.org/10.3103/S1066369X20090029