Schwarz lemma for harmonic functions in the unit ball

Recently, it is proven that positive harmonic functions defined in the unit disc or the upper half-plane in $\mathbb{C}$ are contractions in hyperbolic metrics \cite{Markovic}. Furthermore, the same result does not hold in higher dimensions as shown by given counterexamples \cite{Melentijevic-P}. In this paper, we shall show that positive (or bounded) harmonic functions defined in the unit ball in ${\mathbb{R}}^n$ are Lipschitz in hyperbolic metrics. The involved method in main results allows to establish essential improvements of Schwarz type inequalities for monogenic functions in Clifford analysis \cite{Zhang14,Zhang16} and octonionic analysis \cite{Wang-Bian-Liu} in a unified approach.