The Harnack distance and estimates of subharmonic functions from below on the unit disc

TL;DR

This paper demonstrates that the classic Harnack distance suffices for estimating subharmonic functions from below on the unit disc, simplifying the technical approach in potential theory.

Contribution

It shows that the Harnack distance alone can be used for lower estimates of subharmonic functions, reducing reliance on complex technical tools.

Findings

Harnack distance effectively estimates subharmonic functions from below

Simplifies potential theory techniques for the unit disc

Reduces need for diverse technical apparatus in growth estimates

Abstract

General estimates from below of holomorphic and subharmonic functions play one of the key roles in the theory of growth of holomorphic and subharmonic functions and in general in the theory of potential. At the same time, the most diverse technical apparatus was often used for this. In one of our recent papers, we showed in fairly general cases that in fact such a wide set of tools is not required. The classic Harnack distance is almost always enough. In this article, we will illustrate how this works in the cases of the unit disc in the complex plane.