Extension operators and Janowski starlikeness with complex coefficients

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 [7] and let \(\Phi_{n, Q}\) be the extension operator introduced in [16]. Let \(a \in \mathbb{C}\), \(b \in \mathbb{R}\) be such that \(|1-a| < b \leq {\rm Re}\  a\). We consider the Janowski classes \(S^*(a,b, \mathbb{B})\) and \(\mathcal{A} S^*(a,b, \mathbb{B})\) with complex coefficients introduced in [4]. In the case \(n=1\), we denote \(S^*(a,b, \mathbb{B}^1)\) by \(S^*(a,b)\) and \(\mathcal{A} S^*(a,b, \mathbb{B}^1)\) by \(\mathcal{A} S^*(a,b)\). We shall prove that the following preservation properties concerning the extension operator \(\Phi_{n, \alpha, \beta}\) hold: \(\Phi_{n, \alpha, \beta} (S^*(a,b)) \subseteq S^*(a,b, \mathbb{B})\), \(\Phi_{n, \alpha, \beta} (\mathcal{A} S^*(a,b)) \subseteq \mathcal{A} S^*(a,b, \mathbb{B})\). Also, we prove similar results for the extension operator \(\Phi_{n, Q}\) : \(\Phi_{n, Q}(S^*(a,b)) \subseteq S^*(a,b, \mathbb{B})\),  \(\Phi_{n, Q}(\mathcal{A} S^*(a,b)) \subseteq \mathcal{A} S^*(a,b, \mathbb{B}))\).