On convolutions of slanted half-plane mappings

Abstract

The convolution of convex harmonic univalent functions in the unit disk, unlike analytic functions, may not be convex or even univalent. The main purpose of this work is to develop previous work involving the convolution of convex harmonic functions. Briefly, we obtain under which conditions the convolution of a right half-plane harmonic mapping having a dilatation −z and a slanted half-plane harmonic mapping with β having a dilatation e iμ ρ+z 1+ρz (|ρ| < 1 and μ ∈ R) is univalent and convex in the direction −β. We also provide an example illustrating graphically with the help of Maple to illuminate the result.

Convolutions of Harmonic Mappings Convex in the Horizontal Direction

“…In particular, Dorff [4] and Dorff et al [5] studied the convolution of harmonic univalent mappings in the right half-plane. For some recent investigations involving convolution of harmonic mappings, we refer the reader to [6][7][8][9][10][11][12][13].…”

A new parametric differential operator generalized a class of d'Alembert's equations

“…Nowadays, the fractional, fractal and conformable operators play a major role developing applications in engineering, medical studies including the dynamic of recent pandemic, economic and computer sciences. More applications of this theory is appeared, when some classes of differential and integral operators are extended to the complex plane [7][8][9].…”