Directional Convexity of Combinations of Harmonic Half-Plane and Strip Mappings
Commun. Korean Math. Soc.
Published online May 13, 2021
Subzar Beig and Vaithiyanathan Ravichandran
Government Degree College Uri, Baramulla--193 123, Jammu and Kashmir, NATIONAL INSTITUTE OF TECHNOLOGY TIRUCHIRAPPALLI
Abstract : For k = 1, 2, let fk = hk + gk be normalized harmonic right half-planeor vertical strip mappings. We consider the convex combination ˆf = ηf1 + (1 − η)f2 = ηh1+(1−η)h2+ηg1 + (1 − η)g2 and the combination ˜f = ηh1+(1−η)h2+ηg1 + (1 − η)g2. For real η, the two mappings ˆf and ˜f are the same. We investigate the univalence and directional convexity of ˆf and ˜f for η ∈ C. Some sufficient conditions are found for convexity of the combination ˜f.
Keywords : harmonic mappings; directional convexity; harmonic shear; linear combination; strip mappings
MSC numbers : 31A05; 30C45
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