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

Abstract

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

References in corpus (4)

• A Cardioid Domain and Starlike Functions
• The radius of uniform convexity of Bessel functions
• Radii of Starlikeness and Convexity of Bessel Functions
• On Certain Generalizations of ${\mathcal{S}}^{\ast }(\psi )$