${\mathcal {S}}^{*}(\phi )$ and ${\mathcal {C}}(\phi )$-Radii for Some Special Functions

Abstract

The radius problem explaining the geometric properties of the normalized forms of the special functions has been of special interest among the Geometric function theories. 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 ${\mathcal {C}}(1+\alpha z)$, $0<\alpha \le 1$ 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.