Asymptotic Behaviour for a Class of Subharmonic Functions in a Half Space

TL;DR

This paper establishes growth estimates at infinity for a class of subharmonic functions in the upper half space, generalizing known properties of analytic and harmonic functions.

Contribution

It introduces new asymptotic growth bounds for subharmonic functions in half spaces, extending classical results to a broader function class.

Findings

Subharmonic functions grow slower than a specific asymptotic rate at infinity.

The growth estimates generalize properties of harmonic and analytic functions.

Results apply to functions in higher-dimensional half spaces.

Abstract

A class of subharmonic functions are proved to have the growth estimates $u(x)= o(x_n^{1-\frac{\alpha}{p}}|x|^{\frac{\gamma}{p}+\frac{n-1}{q}-n+\frac{\alpha}{p}})$ at infinity in the upper half space of ${\bf R}^{n}$, which generalizes the growth properties of analytic functions and harmonic functions.