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

Pardeep Kaur, Sukhwinder Singh Billing

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Keywords:

analytic function, univalent function, close-to-convex function

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 $\phi.$

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Published

2022-06-30

How to Cite

Team, S. (2022). A Class of Close-to-Convex Functions Satisfying a Differential Inequality: Pardeep Kaur, Sukhwinder Singh Billing. Thai Journal of Mathematics, 20(2), 557–562. Retrieved from https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1344

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