Convolutions of Harmonic Half-Plane Mappings with Harmonic Vertical Strip Mappings

Abstract

In the present paper, we prove the convolutions of generalized harmonic right half-plane mappings with harmonic vertical strip mappings are univalent and convex in the horizontal direction. Furthermore, we use a simple method to prove the Theorem~\ref{thmD} that has been proved Kumar~\cite[Theorem 2.4]{Kumar2}. Finally, some examples of harmonic univalent mappings convex in the horizontal direction are also constructed to illuminate our main results.