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
Mots-clés :
Liouville theorem.
B. N. Khabibullin. Liouville-type theorems for functions of finite order. Ufa mathematical journal, Tome 12 (2020) no. 4, pp. 114-118. http://geodesic.mathdoc.fr/item/UFA_2020_12_4_a9/
@article{UFA_2020_12_4_a9,
author = {B. N. Khabibullin},
title = {Liouville-type theorems for functions of finite order},
journal = {Ufa mathematical journal},
pages = {114--118},
year = {2020},
volume = {12},
number = {4},
language = {en},
url = {http://geodesic.mathdoc.fr/item/UFA_2020_12_4_a9/}
}
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AU - B. N. Khabibullin
TI - Liouville-type theorems for functions of finite order
JO - Ufa mathematical journal
PY - 2020
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UR - http://geodesic.mathdoc.fr/item/UFA_2020_12_4_a9/
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