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A Strengthening of the Harnack Inequality

Marek Svetlik

TL;DR

This paper addresses sharpening Harnack's inequality for positive harmonic functions $u:\mathbb{U}\to(0,\infty)$ on the unit disk. It introduces a stronger bound $\left(\frac{1+|z|^2}{1-|z|^2}+\frac{|\nabla u(0)|}{u(0)}\frac{|z|}{1-|z|^2}\right)^{-1} \le \frac{u(z)}{u(0)} \le \left(\frac{1+|z|^2}{1-|z|^2}+\frac{|\nabla u(0)|}{u(0)}\frac{|z|}{1-|z|^2}\right)$ for all $z\in\mathbb{U}$, derived via the half-plane method and Beardon–Carne's refinement of Schwarz–Pick. The proof constructs a holomorphic lift $f$ with $\mathrm{Re}\,f=u$ and uses hyperbolic distance bounds to translate a bound on $d_{\mathbb{K}}(f(z),f(0))$ into the stated bound on $u(z)/u(0)$. The results include sharpness examples and clarify the role of the gradient at the origin in tightening the classical Harnack inequality.

Abstract

We prove the stronger version of Harnack's inequality for positive harmonic functions defined on the unit disc.

A Strengthening of the Harnack Inequality

TL;DR

This paper addresses sharpening Harnack's inequality for positive harmonic functions on the unit disk. It introduces a stronger bound for all , derived via the half-plane method and Beardon–Carne's refinement of Schwarz–Pick. The proof constructs a holomorphic lift with and uses hyperbolic distance bounds to translate a bound on into the stated bound on . The results include sharpness examples and clarify the role of the gradient at the origin in tightening the classical Harnack inequality.

Abstract

We prove the stronger version of Harnack's inequality for positive harmonic functions defined on the unit disc.
Paper Structure (2 sections, 6 theorems, 35 equations)

This paper contains 2 sections, 6 theorems, 35 equations.

Table of Contents

  1. Introduction and the main result
  2. Proof of the main result

Key Result

Theorem 1

Let $u:\mathbb{U}\rightarrow (0,+\infty)$ be a harmonic function. Then for all $z\in\mathbb{U}$. Here $\nabla u=(u_{x},u_{y})$ is the gradient of the function $u$.

Theorems & Definitions (12)

  • Theorem 1
  • Example 1
  • Theorem 2: BeardonCarne
  • Proposition 1
  • Theorem 3: MMar
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 2 more