Radius of Ma--Minda Starlikeness and Convexity for Functions with Bounded Second Derivative

Abstract

A normalized analytic function $f(z)=z+\sum_{k=n+1}^{\infty}a_kz^k$ defined on the unit disk $\mathbb{D}$ is called Ma--Minda starlike or Ma--Minda convex if $zf'(z)/f(z)$ or $1+zf''(z)/f'(z)$, respectively, is subordinate to a Ma--Minda function $φ$. In this paper, we investigate the radii of Ma--Minda starlikeness and convexity for functions satisfying $|f''(z)|<M$ in $\mathbb{D}$. We first develop a unified approach for determining these radii and then apply it to obtain explicit radius estimates for several important choices of the Ma--Minda function~$φ$.