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

Suppose $ \alpha >-1 $ α > − 1 and $ 1\leq p \leq \infty $ 1 ≤ p ≤ ∞. Let $ f=\mathcal {P}_{\alpha }[F] $ f = P α [ F ] be an α-harmonic mapping on $ {\mathbb D} $ D with the boundary F being absolute continuous and $ \dot {F}\in L^p(0,2\pi) $ F ˙ ∈ L p (0 , 2 π) , where $ \dot {F}({\rm e}^{i\theta }):=\frac {{\rm d}}{{\rm d}\theta }F({\rm e}^{i\theta }) $ F ˙ (e iθ) := d d θ F (e iθ). In this paper, we investigate the membership of $ f_z $ f z and $ f_{\bar {z}} $ f z ¯ in the space $ \mathcal {H}_{\mathcal {G}}^{p}(\mathbb {D}) $ H G p (D) , the generalized Hardy space. We prove, if $ \alpha >0 $ α > 0 , then both $ f_z $ f z and $ f_{\bar {z}} $ f z ¯ are in $ \mathcal {H}_{\mathcal {G}}^{p}(\mathbb {D}) $ H G p (D). If $ \alpha α < 0 , then $ f_z $ f z and $ f_{\bar {z}}\in \mathcal {H}_{\mathcal {G}}^{p}(\mathbb {D}) $ f z ¯ ∈ H G p (D) if and only if f is analytic. Finally, we investigate a Schwarz Lemma for α-harmonic functions. [ABSTRACT FROM AUTHOR]