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 $q(z)$.