Convexity of integral transforms and function spaces

For β<1, let Pγ(β) denote the class of all normalized analytic functions f in the unit disc Δ such that for some φ ε ℝ. Let script S sign*(μ), 0 ≤ μ<1, denote the usual class of starlike functions of order μ. Define script K sign (μ)={f: zf′(z) ε script S sign *(μ)}, the class of all convex functions of order μ. In this paper, we consider integral transforms of the form The aim of this paper is to find conditions on λ (t) so that each of the transformations carries script P signγ(β) into script S sign *(μ) or script K sign(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.