On Harnack inequality and harmonic Schwarz lemma

In this paper, we study the \((s, C(s))\)-Harnack inequality in a domain \(G\subset \mathbb{R}^n\) for \(s\in(0,1)\) and \(C(s)\geq1\) 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 \(K\geq 1\). Finally, we provide a type of harmonic Schwarz lemma and improve the Schwarz-Pick estimate for a real-valued harmonic function.