Sharp Pointwise Estimate of $$\alpha $$ -Harmonic Functions

Abstract

Let \(\alpha >-1\) and assume that f is \(\alpha \)-harmonic mapping defined in the unit disk that belongs to the Hardy class \(h^p\) with \(p\geqslant 1\). We obtain some sharp estimates of the type \(|f(z)|\le g(|r|) \Vert f^*\Vert _p\) and \(|Df(z)|\le h(|r|)\Vert f^*\Vert _p\). We also prove a Schwarz type lemma for the class of \(\alpha \)-harmonic mappings of the unit disk onto itself fixing the origin.