On certain classes of close-to-convex functions

Abstract

A function f, analytic in the unit disk E and given by , f(z)=z+∑k=2∞anzk is said to be in the family Kn if and only if Dnf is close-to-convex, where Dnf=z(1−z)n+1∗f, n∈N0={0,1,2,…} and ∗ denotes the Hadamard product or convolution. The classes Kn are investigated and some properties are given. It is shown that Kn+1⫅Kn and Kn consists entirely of univalent functions. Some closure properties of integral operators defined on Kn are given.

Keywords

• univalent
• close-to-convex
• starlike
• convolution
• integral operators.