Convexity of integral transforms and function spaces

Abstract

For β<1, let P_γ(β) denote the class of all normalized analytic functions f in the unit disc Δ 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 starlike functions of order μ. Define K(μ)={f: zf'(z)∈S^∗(μ)}, the class of all convex functions of order μ. In this paper, we consider integral transforms of the form V_λ (f)(z) = ∫₀¹λ(t) f(tz)/t dt and ν_λ(f)(z) = z ∫₀¹λ(t) 1-ρtz/1-tz dt ∗ f(z) . The aim of this paper is to find conditions on λ(t) so that each of the transformations carries P_γ(β) into S^∗(μ) or K(μ). A number of applications for certain special choices of λ(t) are also established. These results extend the previously known results by a number of authors.