A Generalized Class of Close-to-Convex Functions

Pardeep Kaur, Sukhwinder Singh Billing

Let $\mathcal{H}^{\phi}_{\alpha}(\beta)$ denote the class of functions $f,$ analytic in the open unit disk $\mathbb E$ which satisfy the condition$$\Reeft((1-lpha)\frac{zf'(z)}{hi(z)}+lphaeft(1+\frac{zf''(z)}{f'(z)}\right)\right)>\beta,\quad z ı\mathbb{E}, $$ where $\alpha,~\beta$ are pre-assigned real numbers and $\phi(z)$ is a starlike function. The special cases of the class $\mathcal{H}^{\phi}_{\alpha}(\beta)$ have been studied in literature by different authors. In 2007, Singh et al. \cite{singhs2007} studied the class $\mathcal{H}^{z}_{\alpha}(\beta)$ and they established that functions in $\mathcal{H}_{\alpha}^{z}(\beta)$ are univalent for all real numbers $\alpha, ~\beta$ satisfying the condition $\alpha\leq\beta<1$ and the result is sharp in the sense that constant $\beta$ cannot be replaced by a real number smaller than $\alpha.$ Singh et al. \cite{singhv2005} in 2005, proved that for $0<\alpha<1$ functions in class $\mathcal{H}_{\alpha}^{z}(\alpha)$ are univalent. In 1975, Al-Amiri and Reade \cite{alamiri} showed that functions in class $\mathcal{H}_{\alpha}^{z}(0)$ are univalent for all $\alpha\leq 0$ and also for $\alpha=1$ 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 $\alpha,~ \beta$ and for a starlike function $\phi$ satisfying the condition $\beta+\alpha-1<\alpha \Re\left(\frac{z\phi'(z)}{\phi(z)}\right)\leq\beta<1$.