Lower bounds for subharmonic functions in terms of the Harnack distance

TL;DR

This paper establishes new lower bounds for subharmonic functions within bounded domains using the Harnack distance, with novel results for planar domains and intervals, impacting various classes of functions.

Contribution

It introduces general estimates from below for subharmonic functions based on the Harnack distance, including new results for planar domains and intervals.

Findings

New lower bounds for subharmonic functions in bounded domains.

Harnack distance is crucial for these estimates.

Applications to various classes of functions are discussed.

Abstract

Let $G$ be a nonempty bounded domain in a finite-dimensional Euclidean space. The main results are general estimates from below at points from $G$ for an arbitrary subharmonic function $u\not\equiv -\infty$ on the closure of the domain $G$ through the maximum of the function $u$ on the boundary of the domain $G$. These results are new for planar domains $G$, and for intervals of $G$ on the numerical line have also not been previously noted. They show that the Harnack distance plays a key role in these estimates. Further applications to subharmonic, convex, holomorphic functions, as well as to meromorphic functions and differences of subharmonic functions in domains of a particular type are supposed to be outlined in the continuation of this article.