Polynomial growth of subharmonic functions in a strongly symmetric Riemannian manifold

TL;DR

This paper investigates the properties of subharmonic functions in strongly symmetric Riemannian manifolds, introducing a generalized notion of polynomial growth and establishing key bounds and growth degrees.

Contribution

It introduces a new concept of polynomial growth of functions relative to a real function and proves that non-negative twice differentiable subharmonic functions exhibit polynomial growth of degree 1.

Findings

Non-negative twice differentiable subharmonic functions have polynomial growth of degree 1.

A lower bound for the integral of convex functions over geodesic balls.

Generalization of polynomial growth concept in Riemannian manifolds.

Abstract

In this article we have studied some properties of subharmonic functions in a strongly symmetric Riemannian manifold with a pole. As a generalization of polynomial growth of a function we have introduced the notion of polynomial growth of some degree of a function with respect to a real function and proved that any non-negative twice differentiable subharmonic functions in an $n$-dimensional manifold always admit polynomial growth of degree $1$ with respect to a non-negative real valued subharmonic function on real line. We have also given a lower bound of the integration of a convex function in a geodesic ball.