Inclusion properties of certain subclasses of analytic functions defined by generalized Salagean operator

Abstract

Let $A$ denote the class of analytic functions with the normalization $f(0)=f^{\prime }(0)-1=0$ in the open unit disc $U=\{z:\left\vert z\right\vert <1\}$.  Set $$f_{\lambda }^{n}(z)=z+\sum_{k=2}^{\infty }[1+\lambda (k-1)]^{n}z^{k}\quad(n\in N_{0};\ \lambda \geq 0;\ z\in U),$$ and define $f_{\lambda ,\mu }^{n}$ in terms of the Hadamard product $$f_{\lambda }^{n}(z)\ast f_{\lambda ,\mu }^{n}=\frac{z}{(1-z)^{\mu }}\quad (\mu >0;\ z\in U).$$ In this paper, we introduce several subclasses of analytic functions defined by means of the operator $I_{\lambda ,\mu }^{n}:A\longrightarrow A$, given by $$I_{\lambda ,\mu }^{n}f(z)=f_{\lambda ,\mu }^{n}(z)\ast f(z)\quad (f\in A;\ n\in N_{0;}\ \lambda \geq 0;\ \mu >0).$$ Inclusion properties of these classes and the classes involving the generalized Libera integral operator are also considered.