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 $p\ge 1$. We obtain some sharp estimates of the type $|f(z)|\le g(|r|)\Vert f^{\ast }{\Vert }_p$ and $|Df(z)|\le h(|r|)\Vert f^{\ast }{\Vert }_p$. We also prove a Schwarz type lemma for the class of $\alpha -$harmonic mappings of the unit disk onto itself fixing the origin.

12 pages