Opuscula Math. 33, no. 4 (2013), 697-700
http://dx.doi.org/10.7494/OpMath.2013.33.4.697

Opuscula Mathematica

# A note on bounded harmonic functions over homogeneous trees

Francisco Javier González Vieli

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.

Keywords: bounded harmonic function, homogeneous tree.

Mathematics Subject Classification: 31C20, 05C05, 05C63.

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• Francisco Javier González Vieli
• Montoie 45, 1007 Lausanne, Switzerland