Sufficiency for nephroid starlikeness using hypergeometric functions

Let A consists of analytic functions f:D→C satisfying f(0)=f′(0)−1=0. Let SNe∗ be the recently introduced Ma–Minda type functions family associated with the two‐cusped kidney‐shaped nephroid curve (u−1)2+v2−493−4v23=0 given by SNe∗:=f∈A:zf′(z)f(z)≺φNe(z)=1+z−z3/3. In this paper, we adopt a novel technique that uses the geometric properties of hypergeometric functions to determine sharp estimates on β so that each of the differential subordinations p(z)+βzp′(z)≺1+z;1+z;ez; imply p (z) ≺ φNe(z), where p (z) is analytic satisfying p(0)=1. As applications, we establish conditions that are sufficient to deduce that f∈A is a member of SNe∗.