Extension operators and Janowski starlikeness with complex coe cients

Abstract

"In this paper, we obtain certain generalizations of some results from [13] and [14]. Let $\Phi_{n, \alpha, \beta}$ be the extension operator introduced in \cite{GrahamHamadaKohrSuffridge} and let $\Phi_{n, Q}$ be the extension operator introduced in [7]. Let $a \in \C$, $b \in \R$ be such that $|1-a| < b \leq {\rm Re}\ a$. We consider the Janowski classes $S^*(a,b, \B)$ and $\A S^*(a,b, \B)$ with complex coefficients introduced in [16]. In the case $n=1$, we denote $S^*(a,b, \mathbb{B}^1)$ by $S^*(a,b)$ and $\A S^*(a,b, \mathbb{B}^1)$ by $\A S^*(a,b)$. We shall prove that the following preservation properties concerning the extension operator $\Phi_{n, \alpha, \beta}$ hold: $\Phi_{n, \alpha, \beta} (S^*(a,b)) \subseteq S^*(a,b, \B)$, $\Phi_{n, \alpha, \beta} (\A S^*(a,b)) \subseteq \A S^*(a,b, \B)$. Also, we prove similar results for the extension operator $\Phi_{n, Q}$: $$\Phi_{n, Q}(S^*(a,b)) \subseteq S^*(a,b, \B),\ \Phi_{n, Q}(\A S^*(a,b)) \subseteq \A S^*(a,b, \B).$$ "