Radius problems for functions associated with a nephroid domain

Abstract

Let ${\mathcal {S}}^*_{Ne}$ be the collection of all analytic functions f(z) defined on the open unit disk ${\mathbb {D}}$ and satisfying the normalizations $f(0)=f'(0)-1=0$ such that the quantity $zf'(z)/f(z)$ assumes values from the range of the function $\varphi _{\scriptscriptstyle {Ne}}(z):=1+z-z^3/3,z\in {\mathbb {D}}$, which is the interior of the nephroid given by

$$\begin{aligned} \left( (u-1)^2+v^2-\frac{4}{9}\right) ^3-\frac{4 v^2}{3}=0. \end{aligned}$$

In this work, we find sharp ${\mathcal {S}}^*_{Ne}$-radii for several geometrically defined function classes introduced in the recent past. In particular, ${\mathcal {S}}^*_{Ne}$-radius for the starlike class ${\mathcal {S}}^*$ is found to be 1/4. Moreover, radii problems related to the families defined in terms of ratio of functions are also discussed. Sharpness of certain radii estimates are illustrated graphically.