On Harnack inequality and harmonic Schwarz lemma

Abstract

In this paper, we study the (s, C(s))-Harnack inequality in a domain G ⊂ Rn for s ∈ (0, 1) and C(s) ≥ 1 and present a series of inequalities related to (s, C(s))-Harnack functions and the Harnack metric. We also investigate the behavior of the Harnack metric under K-quasiconformal and K-quasiregular mappings, where K ≥ 1. Finally, we provide a type of harmonic Schwarz lemma and improve the Schwarz-Pick estimate for a real-valued harmonic function.