Liouville-type theorems with constraints outside of small sets on circles or spheres for functions of finite order

TL;DR

This paper establishes Liouville-type theorems for various classes of functions of finite order, showing that boundedness outside small sets on spheres implies global boundedness and constancy.

Contribution

It introduces new Liouville-type results for subharmonic, entire, plurisubharmonic, convex, and harmonic functions of finite order, extending classical theorems to broader contexts.

Findings

Subharmonic functions of finite order bounded outside small sets are globally bounded.

Entire and plurisubharmonic functions of finite order are constant if bounded outside small sets.

Results are also new for functions of one complex variable.

Abstract

We prove that subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some asymptotically small sets on spheres, 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 of such sets on spheres are constant. Our results and methods of proof are also new for functions of one complex variable.