On convex combinations of harmonic mappings

Abstract

Abstract Let ψ μ , ν ( z ) = ( 1 − 2 cos ν e i μ z + e 2 i μ z 2 ) − 1 $\psi_{\mu,\nu}(z)=(1-2\cos\nu e^{i\mu}z+e^{2i\mu}z^{2})^{-1}$ , μ , ν ∈ [ 0 , 2 π ) $\mu,\nu\in[0,2\pi)$ and p be an analytic mapping with Re p > 0 $\operatorname{Re} p>0$ on the open unit disk. We consider the sense-preserving planar harmonic mappings f = h + g ‾ $f=h+\overline{g}$ , which are shears of the mapping ∫ 0 z ψ μ , ν ( ξ ) p ( ξ ) d ξ $\int_{0}^{z} \psi_{\mu,\nu}(\xi) p(\xi)\,{d}\xi$ in the direction μ. These mappings include the harmonic right half-plan mappings, vertical strip mappings, and their rotations. For various choices of dilatations g ′ / h ′ $g'/h'$ of f, sufficient conditions are found for the convex combinations of these mappings to be univalent and convex in the direction μ.

Keywords

• Harmonic mappings
• Convex combination
• Directional convexity