Duality techniques for certain integral transforms to be starlike ✩

For β <1, let Pγ (β) denote the class of all normalized analytic functions f such that Re eiϕ (1 − γ ) f (z) z + γf (z)− β > 0, z∈ Δ, for some ϕ ∈ R. Let S∗(μ), 0 μ<1, denote the usual class of starlike functions of order μ, S∗(μ) = f: Re zf (z) f (z) >μ, z ∈ Δ . In this paper we find conditions so that the integral transform Vλ(f )(z) = 1 0 λ(t) f (tz) t dt carries Pγ (β) into S∗(μ). We provide a number of applications for certain special choices of λ(t). These results extend previously known results by a number of authors.  2004 Elsevier Inc. All rights reserved