Duality techniques for certain integral transforms to be starlike

Abstract

For β<1, let P_γ(β) denote the class of all normalized analytic functions f such that Re{e^iφ (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 stalike 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 V_λ(f)(z) = ∫₀¹λ(t) f(tz)/t dt carries P_γ(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.