Coefficient inequalities for certain starlike and convex functions

Abstract

In this paper, we consider two Ma--Minda-type subclasses of starlike and convex functions associated with the normalized analytic function $\varphi_{Ne}(z)=1+z-z^3/3$ that maps an open unit disk onto the Nephroid shaped bounded domain in the right--half of the complex plane. We investigate convolution and quasi-Hadamard product properties for the functions belonging to such classes. In addition, we compute best possible estimates on third order Hermitian--Toeplitz determinant and non-sharp estimates on certain third order Hankel determinants for the starlike functions associated with the interior region of Nephroid.