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Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions. (English) Zbl 1182.30017

Summary: Let 𝒰(λ,μ) denote the class of all normalized analytic functions f in the unit disk |z|<1 satisfying the condition

f(z) z0andf ' (z)z f(z) μ+1 -1<λ,z<1·

For f𝒰(λ,μ) with μ1 and 0μ 1 1, and for a positive real-valued integrable function ϕ defined on [0,1] satisfying the normalizing condition 0 1 ϕ(t)dt=1, we consider the transform G ϕ f(z) defined by

G ϕ f(z)=z 0 1 ϕ(t)zt f(tz) μ dt -1/μ 1 ,zΔ·

In this paper, we find conditions on the range of the parameters λ and μ so that the transform G ϕ f is univalent or starlike. In addition, for a given univalent function of a certain form, we provide a method for obtaining functions in the class 𝒰(λ,μ).

MSC:
30C45Special classes of univalent and multivalent functions
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