Schwarz–Pick Lemma for Harmonic and Hyperbolic Harmonic Functions

We establish some inequalities of Schwarz–Pick type for harmonic and hyperbolic harmonic functions on the unit ball of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}^n}$$\end{document} and we disprove a recent conjecture of Liu (Int Math Res Not, 2021).