Sharp pointwise estimate of $α-$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 $|f(z)|\le g(|r|) |f^\ast|_p$ and $|Df(z)|\le h(|r|)|f^\ast|_p$. We also prove a Schwarz type lemma for the class of $\alpha-$harmonic mappings of the unit disk onto itself fixing the origin.