A Class of Close-to-Convex Functions Satisfying a Differential Inequality

Pardeep Kaur, Sukhwinder Singh Billing

Abstract

Let $\mathcal{H}^{\phi}_{\alpha}(\beta)$ denote the class of functions $f,$ analytic in the open unit disk $\mathbb E,$ which satisfy the condition \[\Re\left[(1-\alpha)\frac{zf'(z)}{\phi(z)}+\alpha\left(2+\frac{zf''(z)}{f'(z)}-\frac{z\phi'(z)}{\phi(z)}\right)\right]>\beta,~z\in\mathbb{E}, \] where $\alpha,~\beta$ are pre-assigned real numbers and $\phi$ is a starlike function in $\mathbb{E}.$ In the present paper, we prove that members of the class $\mathcal{H}^{\phi}_{\alpha}(\beta)$ are close-to-convex and hence univalent for real numbers $\alpha,~ \beta,~\alpha\leq\beta<1$ and for a starlike function $\phi.$