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.
Categories: math.CV