  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
$\mathcal{S}^*_{Ne}$.
