$\mathcal{S}^{*}(φ)$ and $\mathcal{C}(φ)$-radii for some special functions

Abstract: In this paper, we consider the Ma-Minda classes of analytic functions $\mathcal{S}^{*}(\phi):= {f\in \mathcal{A} : ({zf'(z)}/{f(z)}) \prec \phi(z) }$ and $\mathcal{C}(\phi):= {f\in \mathcal{A} : (1+{zf''(z)}/{f'(z)}) \prec \phi(z) }$ defined on the unit disk $\mathbb{D}$ and show that the classes $\mathcal{S}^{*}(1+\alpha z)$ and $\mathcal{C}(1+\alpha z)$, $0<\alpha \leq 1$ solve the problem of finding the sharp $\mathcal{S}^{*}(\phi)$-radii and $\mathcal{C}(\phi)$-radii for some normalized special functions, whenever $\phi(-1)=1-\alpha$. Radius of strongly starlikeness is also considered.