Linear combinations of harmonic quasiconformal mappings convex in one direction

Abstract

In this paper, we introduce a new class $\mathcal{S}_{H}$ (k, γ; φ) of harmonic quasiconformal mappings, where k ∈ [0,1), γ ∈ [0,π) and φ is an analytic function. Sufficient conditions for the linear combinations of mappings in such classes to be in a similar class, and convex in a given direction, are established. In particular, we prove that the images of linear combinations in this class, for special choices of γ and φ, are convex.