Integral operators of certain univalent functions

Om P Ahuja

doi:10.1155/s0161171285000710

Abstract

A function f, analytic in the unit disc Δ, is said to be in the family Rn(α) if for some α(0 ≤ α < 1) and for all z in Δ, where n ϵ No, No = {0, 1, 2, …}. The The class Rn(α) contains the starlike functions of order α for n ≥ 0 and the convex functions of order α for n ≥ 1. We study a class of integral operators defined on Rn(α). Finally an argument theorem is proved.

Highlights

Department of Mathematics University of Papua New GuineaA function f, analytic in the unit disc A, is said to be in the (n)} family R ((z) if Re{(znf(z))(n+|)/(zn-;f(z))
Let A d,note the family of functions f which are analytic in the unit disc Iz {z"< I} and normalised such that f(0) 0 f’(0) I
We study a class of integral operators defined on R (a)

Summary

Department of Mathematics University of Papua New Guinea

A function f, analytic in the unit disc A, is said to be in the (n)} family R ((z) if Re{(znf(z))(n+|)/(zn-;f(z)). The class R (a) contains the starlike functions of order a for n 0 and the convex functions of order We study a class of integral operators defined on R (a). KEY WOROS ANO PHRASES: Univalent, convolution, starlike, convex 1980 AM5 SUBJECT CLASSIFICATION CODES" Primary 50C45, 50C99; Secondary 50C55

We have

Then the function F can also we written in the form

Using the well known estimates

Vn c