Some results on certain subclasses of analytic functions involving generalized hypergeometric functions and Hadamard product

Khalida Inayat Noor

doi:10.1155/s0161171292000164

Abstract

By using a certain linear operator defined by a Hadamard product or convolution, several interesting subclasses of analytic functions in the unit disc are introduced and some unifying relationships between them are established. A variety of characterization results involving a certain functional and some general functions of hypergeometric type are investigated for these classes.

Highlights

Let A denote the class of the function f of the form f(z)-z + a,z" (1.1)which are analytic in the unit disc EIzl < A function leA is said to be in the class R()if, for z e E and 1 > -1, Re zf (z f(z) > -15a function fA is said to belong to the class V(I) if, for z e E and 1 > -1, -" Re (zf’(z))’ f’(z)’" "We note that [(z)n(f) zf’(z) v(), (1.2)
Which are analytic in the unit disc E
A function fe.A is said to be in the classR(a,c;) if L (a, c )f belongs to R(IB) ffw 15 > -1, and fe V(a,c;f3) if, and only if, zf’ eR(a,c;f3) for > -1

Summary

Introduction

A function fA is said to belong to the class V(I) if, for z e E and 1 > -1, The classes V() and R([) of analytic functions have been defined and studied in [9]. The generalized hypergeometric function ,F, is defined by We define the function ((a, c) by Corresponding to the function (a,c), Carlson and Shaffer [2] defined a linear operator L(a,c)on A by the convolution

Results

Conclusion