  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.

Complex Variables