Abstract
Let be the class of normalized analytic functions in the unit disk and define the class P(β)={f∈A:∃φ∈R such that Re[ e i φ ( f ′ (z)−β)]>0,z∈U}. In this paper we find conditions on the number β and the non-negative weight function λ(t) such that the integral transform V λ (f)(z)= ∫ 0 1 λ(t) f ( t z ) t dt is convex of order γ (0≤γ≤1/2) when f∈P(β). Some interesting further consequences are also considered.MSC:30C45, 33C50.
Highlights
Introduction and definitions LetA denote the class of functions of the form ∞f (z) = z + anzn ( . ) n=which are analytic in the open unit disk U = {z ∈ C : |z| < }
Let A be the class of normalized analytic functions in the unit disk U and define the class P(β) = {f ∈ A : ∃φ ∈ R such that Re[eiφ(f (z) – β)] > 0, z ∈ U }
In this paper we find conditions on the number β and the non-negative weight function λ(t) such that the integral transform Vλ(f )(z) =
Summary
Let A be the class of normalized analytic functions in the unit disk U and define the class P(β) = {f ∈ A : ∃φ ∈ R such that Re[eiφ(f (z) – β)] > 0, z ∈ U }. In this paper we find conditions on the number β and the non-negative weight function λ(t) such that the integral transform Vλ(f )(z) = 1 Introduction and definitions Let A denote the class of functions of the form ) contains as special cases several of the known linear integral or differential operators studied by a number of authors.
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