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 $f(0)=f^{\prime }(0)-1=0$. In this paper, we consider the function ${\phi }_{Ne}(z):=1+z-z^3/3$, which maps the unit circle $\{z:|z|=1\}$ onto a $2$-cusped curve called nephroid given by ${\left((u-1{)}^2+v^2-\frac{4}{9}\right)}^3-\frac{4v^2}{3}=0$, and the function class ${\mathcal{S}}_{Ne}^{\ast }$ defined as \begin{align*} \mathcal{S}^*_{Ne}:=\left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)}\precφ_{\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+β\frac{zp'(z)}{p^j(z)}\prec\mathcal{P}(z), \quad j=0,1,2 \end{align*} implies $p\prec {\phi }_{Ne}$, where $\mathcal{P}(z)$ is certain Carathéodory 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 $p\prec {\phi }_{Ne}$ whenever $p(z)+\beta z{p}^{\prime }(z)\prec \sqrt{1+z}$ or $1+z$. As applications, we establish sufficient conditions for $f\in \mathcal{A}$ to be in the class ${\mathcal{S}}_{Ne}^{\ast }$.

21 pages, 11 figures