On harmonic functions and the hyperbolic metric

TL;DR

This paper proves that positive harmonic functions in the upper half-plane act as contractions under hyperbolic metrics, extending understanding of their geometric properties and connections to hyperbolic geometry.

Contribution

It establishes new contraction properties of positive harmonic functions with respect to hyperbolic metrics in the upper half-plane.

Findings

Positive harmonic functions are contractions in hyperbolic metric of the upper half-plane.

Harmonic functions also contract hyperbolic metrics on the positive real line.

Results extend previous work by Kalaj and Vuorinen.

Abstract

Motivated by some recent results of Kalaj and Vuorinen (Proc. Amer. Math. Soc., 2012), we prove that positive harmonic functions defined in the upper half--plane are contractions w.r.t. hyperbolic metrics of half--plane and positive part of the real line, respectively