A note on bounded harmonic functions over homogeneous trees

Abstract

Let $\mathcal{T}_k$ be the homogeneous tree of degree $k\geq 3$. J. M. Cohen and F. Colonna have proved that if $f$ is a bounded harmonic function on $\mathcal{T}_k$, then $|f(x)-f(y)|\leq \|f\|_\infty\cdot 2(k-2)/k$ for any adjacent vertices $x$ and $y$ in $\mathcal{T}_k$. We give here a new and very simple proof of this inequality.