Abstract
Some inclusion and convolution properties of certain subclasses of meromorphic functions associated with a family of multiplier transformations, which are defined by means of the Hadamard product (or convolution), are investigated. We also obtain closure properties for certain integral operators. MSC:30C45, 30C80.
Highlights
Let A denote the class of analytic functions f in the open unit disk U = {z ∈ C : |z| < } with the usual normalization f ( ) = f ( ) – =
If f and g are analytic in U, we say that f is subordinate to g in U, written as f ≺ g or f (z) ≺ g(z), if there exists a Schwarz function w such that f (z) = g(w(z)) (z ∈ U)
A function f ∈ A is said to be prestarlike of order α in U if
Summary
Let N be the class of all functions h which are analytic and univalent in U and for which h(U) is convex with h( ) =. By using the operator Iλn,μ, we introduce the following class of analytic functions for γ > , λ > , s ∈ R, μ > and h ∈ N : Mnλ,μ(γ ; h) := f ∈ M : ( – γ )zIλn,μf (z) + γ z Iλn,μf (z) ≺ h(z). ) that ( + γ )zIλn,μ f (z) + γ z Iλn,μ f (z) h(xz) dμ(x) ≺ h(z), which completes the proof of Theorem. Yields z g(z) ≺ h(z) = μz–μ tμ– h(t) dt ≺ h(z), which shows that f ∈ Mnλ,μ+ (γ ; h) ⊂ Mnλ,μ(γ ; h).
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