On meromorphic $ \alpha $-close-to-convex function

Abstract

Let $ B(\alpha) $ denote the class of all functions $ f $ meromorphic in the unit disc $ E $ with $ z f(z) \ne 0 $, $ z^2 f'(z) \ne 0 $ in $ E $ satisfying the condition

$$ \int_{\theta_1}^{\theta_2} Re \left{ \alpha (1+z \frac{f''(z) }{f'(z)} +(1- \alpha) z \frac{f'(z) }{f(z)} \right} d \theta < \pi $$

where $ 0 \le \theta_1 < \theta_2 \le \theta_2 + 2 \pi $, $ z = re^{i \theta} $, $ r < 1 $ and $ \alpha $ is a non-negative real numbers. We call $ f \in B (\alpha) $ a meromorphic $ \alpha $-colse-to-convex function. This paper pertains to the study of some interesting properties of the class $ B (\alpha) $.