Abstract:
We prove that subharmonic or holomorphic functions of finite order on the plane, in space, or on the unit disc or ball that are bounded above on a sequence of circles or spheres, or on a system of embedded discs or balls, outside some asymptotically small sets are bounded above throughout. Hence, subharmonic functions of finite order on the complex plane, entire or plurisubharmonic functions of finite order, and also convex or harmonic functions of finite order that are bounded above on spheres outside such sets are constants. The results and the approaches to the proofs are new for both functions of one and several variables.
Bibliography: 14 titles.
Keywords:
entire function of finite order, (pluri)subharmonic function, holomorphic function in the unit ball, convex function, Liouville's theorem.
This research was carried out with the support of the Ministry of Science and Higher Education of the Russian Federation in the framework of the Programme for the Support of the Development of the Scientific and Educational Mathematical Center of the Volga Federal District (agreement no. 075-02-2021-1393).
Citation:
B. N. Khabibullin, “Global boundedness of functions of finite order that are bounded outside small sets”, Sb. Math., 212:11 (2021), 1615–1625
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\by B.~N.~Khabibullin
\paper Global boundedness of functions of finite order that are bounded outside small sets
\jour Sb. Math.
\yr 2021
\vol 212
\issue 11
\pages 1615--1625
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Linking options:
https://www.mathnet.ru/eng/sm9502
https://doi.org/10.1070/SM9502
https://www.mathnet.ru/eng/sm/v212/i11/p116
This publication is cited in the following 2 articles:
A. Baranov, “Cauchy–de Branges spaces, geometry of their reproducing kernels and multiplication operators”, Milan J. Math., 91:1 (2023), 97
B. N. Khabibullin, “Integrals of a difference of subharmonic functions against measures and the Nevanlinna characteristic”, Sb. Math., 213:5 (2022), 694–733
Citation in format AMSBIB
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