Properties of close-to-convexity preserved by some integral operators

Abstract

Let C(β), S∗(β), and K(β, λ) be the classes of univalent functions defined in E = {z: ¦z¦< 1}, which are convex of order β, starlike of order β and close-to-convex of order β type λ. Let f(z) = ( 1 α )z 1− 1 α ∝ z 0z 1 x−2 F(z)dz, 0 ⩽ α < 1 . We discuss the properties of the function f when this function F belongs to the class K( β, λ) and its various subclasses.