Differential Subordinations for Starlike Functions Associated With A Nephroid Domain

Abstract

Let $\mathcal{A}$ be the set of all analytic functions $f$ defined in the open unit disk $\mathbb{D}$ and satisfying $f(0)=f'(0)-1=0$. In this paper, we consider the function $\varphi_{\scriptscriptstyle {Ne}}(z):=1+z-z^3/3$, which maps the unit circle $\{z:|z|=1\}$ onto a $2$-cusped curve called nephroid given by $\left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0$, and the function class $\mathcal{S}^*_{Ne}$ defined as \begin{align*} \mathcal{S}^*_{Ne}:=\left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)}\prec\varphi_{\scriptscriptstyle {Ne}}(z)\right\}, \end{align*} where $\prec$ denotes subordination. We obtain sharp estimates on $\beta\in\mathbb{R}$ so that the first-order differential subordination \begin{align*} 1+\beta\frac{zp'(z)}{p^j(z)}\prec\mathcal{P}(z), \quad j=0,1,2 \end{align*} implies $p\prec\varphi_{\scriptscriptstyle{Ne}}$, where $\mathcal{P}(z)$ is certain Carath\'{e}odory function with nice geometrical properties and $p(z)$ is analytic satisfying $p(0)=1$. Moreover, we use properties of Gaussian hypergeometric function in order to get the subordination $p\prec\varphi_{\scriptscriptstyle{Ne}}$ whenever $p(z)+\beta zp'(z)\prec\sqrt{1+z}$ or $1+z$. As applications, we establish sufficient conditions for $f\in\mathcal{A}$ to be in the class $\mathcal{S}^*_{Ne}$.