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- W2113122494 abstract "AbstractRecently, Ali et al [2] defined the class W β (α,γ) consisting of functions fwhich satisfyℜe iφ (1−α+2γ)f(z)z+(α−2γ)f ′ (z)+γzf ′′ (z)−β>0,for all z∈ E= {z: |z| <1} and for α,γ≥ 0 and β<1, φ∈ R(the set of reals). For f∈ W β (α,γ),they discussed the convexity of the integral transformV λ (f)(z) :=Z 10 λ(t)f(tz)tdt,where λis a non-negative real-valued integrable function satisfying the conditionZ 10 λ(t)dt= 1. Theaim of present paper is to find conditions on λ(t) such that V λ (f) is convex of order δ(0 ≤ δ≤ 1/2)whenever f∈ W β (α,γ). As applications, we study various choices of λ(t), related to classical integraltransforms. Key Words: Starlike function, Convex function, Hadamard product, Duality.2000 Mathematics Subject Classification: 30C45, 30C80. 1 Introduction Let A denote the class of analytic functions f defined in the open unit disc E= {z: |z| <1} withthe normalization f(0) = f ′ (0) − 1 = 0. Let A 0 = {g: g(z) = f(z)/z,f∈ A}. Let Sbe the subclassof A consisting of univalent functions in E. A function f ∈ Sis said to be starlike or convex, if fmaps Econformally onto the domains, respectively, starlike with respect to the origin and convex. Thegeneralization of these two classes are given by the following analytic characterizations :S" @default.
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- W2113122494 date "2013-05-03" @default.
- W2113122494 modified "2023-09-27" @default.
- W2113122494 title "Order of Convexity of Integral Transforms and Duality" @default.
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