Radius Problems For Functions Associated with a Nephroid Domai

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