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

M. K. Aouf;  A. Shamandy;  A. O. Mostafa;  S. M. Madian — 2010

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

Let A denote the class of analytic functions with the normalization f ( 0 ) = f ' ( 0 ) - 1 = 0 in the open unit disc U = { z : z < 1 } .  Set f λ n ( z ) = z + ∑ k = 2 ∞ [ 1 + λ ( k - 1 ) ] n z k ( n ∈ N 0 ; λ ≥ 0 ; z ∈ U ) , and define f λ , μ n in terms of the Hadamard product f λ n ( z ) * f λ , μ n = z ( 1 - z ) μ ( μ > 0 ; z ∈ U ) . In this paper, we introduce several subclasses of analytic functions defined by means of the operator I λ , μ n : A ⟶ A , given by I λ , μ n f ( z ) = f λ , μ n ( z ) * f ( z ) ( f ∈ A ; n ∈ N 0 ; λ ≥ 0 ; μ > 0 ) . Inclusion properties of these classes and the classes involving the generalized Libera integral operator are also considered.

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