ON CONVEX p-VALENT FUNCTIONS MAPPED ONTO THENEPHROID DOMAIN

Abstract

 p-valent functions serve as natural generalizations of univalent
functions and offer a broad platform for studying geometric and functional properties within complex analysis. While coefficient estimation is a core problem in geometric function theory (GFT), the specific bounds and functional determinants for convex p-valent functions associated with the nephroid domain remain under-explored. This research addresses this gap by investigating a new subclass of functions characterized by subordination to a kidney-shaped region, which is motivated by the need to extend existing univalent results to broader p-valent classes. Using the theory of subordination and Taylor–Maclaurin series expansions, the methodology involves comparing the structural coefficients of
convex p-valent functions against the nephroid-type mapping

P(ξ) = 1+ξ−ξ3/3 . 

The author establishes estimates for the initial coefficients |a1+p| and |a2+p|, and derives a generalized coefficient bound for |an+p|. These findings are verified for consistency by reducing the results to the specific univalent case where p =1.