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Subordination results for spiral-like functions associated with the Srivastava-Attiya operator. (English) Zbl 1244.30015

Let U be the complex unit disc and 𝒜 the class of functions f with f(z)=z+a 2 z 2 +. For such functions, the authors define the integral operator

𝒥 μ,b m,k f(z)=z+ n=2 C n m (b,μ)a n z n ,

where C n m (b,μ)=1+b n+b μ m!(n+k-2)! (k-2)!(n+m-1)!, b{ 0 - }, μ, k2 and m>-1. This is a generalization of other integral operators studied by various authors. Further, for 0λ, γ<1 and -π/2<η<π/2, they define the subclass of 𝒜, denoted by 𝒢 μ,b m,k (η,γ,λ), which consists of functions f satisfying

Re e iη z(𝒥 μ,b m,k f(z)) ' (1-λ)𝒥 μ,b m,k f(z)+λ(𝒥 μ,b m,k f(z)) ' >γcosη

for zU. Some subclasses are also given and the main result provides a sufficient condition for a function f𝒜 to be also in 𝒢 μ,b m,k (η,γ,λ). Other properties of the class 𝒢 μ,b m,k (η,γ,λ) are given in the second theorem and a corollary following from it.

MSC:
30C45Special classes of univalent and multivalent functions