Liouville-type theorems outside of small exceptional sets for functions of finite order

TL;DR

This paper establishes that certain classes of functions of finite order, which are bounded above outside negligible sets, are actually bounded above everywhere, leading to their constancy under various conditions.

Contribution

It proves Liouville-type theorems for functions of finite order outside sets of zero relative Lebesgue density, extending classical results to broader contexts.

Findings

Convex functions of finite order bounded outside negligible sets are globally bounded.

Subharmonic functions of finite order with boundedness outside negligible sets are globally bounded.

Such functions are constant if bounded outside sets of zero relative Lebesgue density.

Abstract

We prove that convex functions of finite order on the real line and subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some set of zero relative Lebesgue density, are bounded from above everywhere. It follows that subharmonic functions of finite order on the complex plane, entire and plurisubharmonic functions of finite order, and convex or harmonic functions of finite order bounded from above outside some set of zero relative Lebesgue density are constant.