Extreme points and support points of harmonic $\alpha$-Bloch mappings

Abstract

We consider the problem of existence of the extreme points and support points of harmonic $\alpha$-Bloch mappings and little harmonic $\alpha$-Bloch mappings. First, we prove a necessary condition for a little harmonic $\alpha$-Bloch mapping to be an extreme point of the unit ball of the normalized little harmonic $\alpha$-Bloch spaces in the unit disk $\mathbb{𝔻}$, and we also show that a harmonic $\alpha$-Bloch unit-valued set consists of several simple closed pairwise disjoint analytic curves and several isolated points. Then we show that a harmonic $\alpha$-Bloch mapping $f$ is a support point of the unit ball of the normalized harmonic $\alpha$-Bloch spaces in $\mathbb{𝔻}$ if and only if the $\alpha$-Bloch unit-valued set of $f$ is not empty. We also give a characterization for the support points of the unit ball of the harmonic $\alpha$-Bloch spaces in $\mathbb{𝔻}$.