Radius Problems for the New Product of Planar Harmonic Mappings

Abstract

Due to the limitations of the harmonic convolution defined by Clunie and Sheil Small (Ann Acad Sci Fenn Ser A I Math 9:3–25, 1984), a new product \(\otimes \) has been recently introduced (2021) for two harmonic functions defined in an open unit disk of the complex plane. In this paper, the radius of univalence (and other radii constants) for the products \(K\otimes K\) and \(L\otimes f\) are computed, where K denotes the harmonic Koebe function, L denotes the harmonic right half-plane mapping and f is a sense-preserving harmonic function defined in the unit disk with certain constraints. In addition, several conditions on harmonic function f are investigated under which the product \(L\otimes f\) is sense-preserving and univalent in the unit disk.