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.
Highlights
Let A denote the class of functions of the form: (1.1)f (z) = z + akzk, k=22000 Mathematics Subject Classification. 30C45
For 0 ≤ η < 1, we denote by S∗(η), K(η) and C the subclasses of A consisting of all analytic functions which are, respectively, starlike of order η, convex of order η and close-to-convex of order η in U
The remaining part of the proof in Theorem 6 is similar to that of Theorem 3 and so we omit it
Summary
Let A denote the class of functions of the form:. 2000 Mathematics Subject Classification. 30C45. Inclusion properties of certain subclasses of analytic functions. By using the operator Iλn,μ, we introduce the following classes of analytic functions for φ, ψ: Sλn,μ (η; φ) = f ∈ A : Iλn,μf (z) ∈ S∗ (η; φ) , Kλn,μ (η; φ) = f ∈ A : Iλn,μf (z) ∈ K (η; φ) and. We investigate several inclusion properties of the classes Sλn,μ (η; φ), Kλn,μ (η; φ) and Cλn,μ(η, δ; φ, ψ) associated with the operator Iλn,μ. Some applications involving these and other classes of integral operators are considered. With the help of Lemma 1, we obtain the following theorem
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