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

A normalized analytic function $f(z)=z+{\sum }_{k=n+1}^{\mathrm{\infty }}{a}_k{z}^k$ defined on the unit disk $\mathbb{D}$ is called Ma–Minda starlike or Ma–Minda convex if $z{f}^{\mathrm{\prime }}(z)\mathrm{/}f(z)$ or $1+z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{f}^{\mathrm{\prime }}(z)$, respectively, is subordinate to a Ma–Minda function $\phi$. In this paper, we investigate the radii of Ma–Minda starlikeness and convexity for functions satisfying $\mathrm{\mid }{f}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{\mid }<M$ 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~$\phi$.