Let the domain $\Omega_{\alpha, \beta}$, $\alpha > 0$,
$-\infty < \beta < 1$, be bounded by a parabola
$y^2 = 4 \alpha (x - \beta)$ in the complex plane $\mathbb{C}$ and let
$P_{\alpha, \beta}$ be the analytic and univalent function with
$P_{\alpha, \beta}(0) = 1$ and
$P_{\alpha, \beta}(\mathcal{U}) = \Omega_{\alpha, \beta}$,
where $\mathcal{U} = \{z : |z| < 1 \}$ denote the
unit disk in the plane. In this paper, we investigate some
interesting properties of a differential subordination of the form
\[ p(z) + \gamma z p^{\prime} (z) \prec P_{\alpha, \beta}(z) \quad (z \in \mathcal{U}) \]
for $\gamma \ge 0$.