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Ufimsk. Mat. Zh., 2020, Volume 12, Issue 4, Pages 117–121 (Mi ufa537)  

Liouville-type theorems for functions of finite order

B. N. Khabibullin

Bashkir State University, Zaki Validi str. 32, 450000, Ufa, Russia

Abstract: A convex, subharmonic or plurisubharmonic function respectively on the real axis, on a finite dimensional real of complex space is called a function of a finite order if it grows not faster than some positive power of the absolute value of the variable as the latter tends to infinity. An entire function on a finite-dimensional complex space is called a function of a finite order if the logarithm of its absolute value is a (pluri-)subharmonic function of a finite order. A measurable set in an $m$-dimensional space is called a set of a zero density with respect to the Lebesgue density if the Lebesgue measure of the part of this set in the ball of a radius $r$ is of order $o(r^m)$ as $r\to +\infty$. In this paper we show that convex function of a finite order on the real axis and subharmonic functions of a finite order on a finite-dimensional real space bounded from above outside some set of a zero relative Lebesgue measure are bounded from above everywhere. This implies that subharmonic functions of a finite order on the complex plane, entire and subharmonic functions of a finite order, as well as convex and harmonic functions of a finite order bounded outside some set of a zero relative Lebesgue measure are constant.

Keywords: entire function, subharmonic function, pluri-subharmonic function, convex function, harmonic function of entire order, Liouville theorem.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 075-02-2020-1421/1
The research is made in the framework of the development program of Scientific and Educational Mathematical Center of Privolzhsky Federal District, additional agreement no. 075-02-2020-1421/1 to agreement no. 075-02-2020-1421.


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English version:
Ufa Mathematical Journal, 2020, 12:4, 114–118 (PDF, 332 kB); https://doi.org/10.13108/2020-12-4-114

Bibliographic databases:

UDC: 517.574 : 517.576 : 517.550.4 : 517.547.2 : 517.518.244
MSC: 32A15, 30D20, 31C10, 31B05, 31A05, 26B25, 26A51
Received: 01.09.2020

Citation: B. N. Khabibullin, “Liouville-type theorems for functions of finite order”, Ufimsk. Mat. Zh., 12:4 (2020), 117–121; Ufa Math. J., 12:4 (2020), 114–118

Citation in format AMSBIB
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\yr 2020
\vol 12
\issue 4
\pages 117--121
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\vol 12
\issue 4
\pages 114--118
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