On a differential subordination for domains bounded by parabolas

Abstract

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 $\gamma \ge 0$.