Norm estimates of the partial derivatives and Schwarz lemma for α-harmonic functions

Abstract

Suppose -1 $ ]]> α > − 1 and 1 ≤ p ≤ ∞ . Let f = P α [ F ] be an α-harmonic mapping on D with the boundary F being absolute continuous and F ˙ ∈ L p ( 0 , 2 π ) , where F ˙ ( e iθ ) := d d θ F ( e iθ ) . In this paper, we investigate the membership of f z and f z ¯ in the space H G p ( D ) , the generalized Hardy space. We prove, if 0 $ ]]> α > 0 , then both f z and f z ¯ are in H G p ( D ) . If α < 0 , then f z and f z ¯ ∈ H G p ( D ) if and only if f is analytic. Finally, we investigate a Schwarz Lemma for α-harmonic functions.