Let $\mathcal{ A}$ consists of analytic functions $f:\mathbb{ D}\to\mathbb{ C}$ satisfying $f(0)=f'(0)-1=0$. Let $\mathcal{ S}^*_{ Ne}$ be the recently introduced Ma-Minda type functions family associated with the $2$-cusped kidney-shaped { \it nephroid} curve $\left((u-1)^2+v^2-\frac{ 4}{ 9}\right)^3-\frac{ 4 v^2}{ 3}=0$ given by \begin{ align*} \mathcal{ S}^*_{ Ne}:= \left\{ f\in\mathcal{ A}:\frac{ zf'(z)}{ f(z)}\prec\varphi_{ \scriptscriptstyle { Ne}}(z)=1+z-z^3/3\right\}. \end{ align*} In this paper, we adopt a novel technique that uses the geometric properties of { \it hypergeometric functions} to determine sharp estimates on $\beta$ so that each of the differential subordinations \begin{ align*} p(z)+\beta zp'(z)\prec \begin{ cases} \sqrt{ 1+z}; 1+z; e^z; \end{ cases} \end{ align*} imply $p(z)\prec\varphi_{ \scriptscriptstyle{ Ne}}(z)$, where $p(z)$ is analytic satisfying $p(0)=1$. As applications, we establish conditions that are sufficient to deduce that $f\in\mathcal{ A}$ is a member of $\mathcal{ S}^*_{ Ne}$.
Categories: math.CV