On certain applications of differential subordinations for $\Phi$-like functions

Abstract

Let $ f(z) $ be a normalized analytic function in $ \Delta=\{z | z\in{\Bbb C} \mbox{ and } |z| <1\} $ satisfying $ f(0)=0 $ and $ f'(0)=1 $. Let $ \Phi $ be an analytic function in a domain containing $ f(\Delta) $, with $ \Phi(0) = 0 $, $ \Phi^{'}(0) = 1 $ and $ \Phi(\omega)\neq 0 $ for $ \omega\in f(\Delta) - \{ 0 \} $. Let $ q(z) $ be a fixed analytic function in $ \Delta$, $q(0)=1 $. The function $ f $ is called $\Phi$-like with respect to $ q $ if

$$ \frac{zf^{'}(z)}{\Phi(f(z))} \prec q(z) \quad (z\in\Delta). $$

In this paper, we obtain some sufficient conditions for functions to be $ \Phi $-like with respect to $ q(z) $.