On a New Subclass of Multivalent Harmonic Mappings Associated with Close-to-Convex Functions

This paper is devoted to defining and analyzing a new subclass M(ν,τ,q) of q-valent harmonic mappings in the unit disk D, as well as investigating its connection with close-to-convex analytic functions. First, we prove that this newly defined class is non-empty and discuss its relationship with several known classes of harmonic mappings. Using arguments similar to those employed in the study of Mocanu-type harmonic mappings, we establish the close-to-convexity of functions belonging to this class. Necessary coefficient estimates for the analytic part are obtained, and auxiliary lemmas which play an essential role in the investigation of geometric properties of the class are derived. In particular, we establish distortion estimates for the derivative of the analytic part, which lead to a growth and distortion theorem for functions in the newly defined class M(ν,τ,q). Furthermore, a covering theorem is obtained for these harmonic mappings. In addition, we also derive sharp bounds for the Fekete–Szegö-type functionals and several graphical examples are presented to analyze the geometric structure of the mappings in the class M(ν,τ,q) that demonstrate how the parameters q,ν, and τ, affect the deformation of the unit disk.