Article
Ufa Mathematical Journal
Volume 12, Number 4, pp. 114-118
Liouville-type theorems for functions of finite order
Khabibullin B.N.
DOI:10.13108/2020-12-4-114
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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.