  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.
