A note on bounded harmonic functions over homogeneous trees

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