Geometric properties and univalency of harmonic functions generated by certain operators

In this work, two operators ψ and ψφ are introduced, which map analytic functions to harmonic functions. For the operator ψ, sufficient conditions for its image to be univalent and close-to-convex, fully (or uniform) starlike (or convex) are given, followed by the study of convolution and radius problems. Meanwhile, the operator ψφ is a generalization of the operator introduced by Ahuja et al. [A technique of constructing planar harmonic mappings and their properties. Kodai Math J. 2017;40(2):278–288. doi: 10.2996/kmj/1499846598], where the sufficient conditions for its image to be univalent and convex in the direction of the real axis, fully starlike (or convex) are given, followed by some sharp coefficient bounds for the images of the Koebe function and the right half-plane function. The work ends with the application of ψ on Mittag-Leffler-Prabhakar function of Le Roy type and Wright function.