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 α-Bloch mappings and little harmonic α-Bloch mappings. First, we prove a necessary condition for a little harmonic α-Bloch mapping to be an extreme point of the unit ball of the normalized little harmonic α-Bloch spaces in the unit disk 𝔻, and we also show that a harmonic α-Bloch unit-valued set consists of several simple closed pairwise disjoint analytic curves and several isolated points. Then we show that a harmonic α-Bloch mapping f is a support point of the unit ball of the normalized harmonic α-Bloch spaces in 𝔻 if and only if the α-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 α-Bloch spaces in 𝔻.