Properties of multivalent functions associated with the integral operator defined by the hypergeometric function

Mohsan Raza,Sarfraz Nawaz Malik

doi:10.1186/1029-242x-2013-458

Abstract

In this paper, we introduce a new class of multivalent functions by using a generalized integral operator defined by the hypergeometric function. Some properties such as inclusion, radius problem and integral preserving are considered. MSC:30C45, 30C50.

Highlights

Introduction and preliminaries LetAp denote the class of functions f (z) of the form ∞f (z) = zp + anzn p ∈ N = {, . . .}, ( . ) n=p+which are analytic in the open unit disc E
If g is univalent in E, we have the following equivalence f (z) ≺ g(z) ⇐⇒ f ( ) = g( ) and f (E) ⊂ g(E)
Respectively, take all the values in the conic domain k,δ defined by k,δ = u + iv : u > k (u – ) + v + δ with p(z) zf (z) f (z) or p(z) zf f (z) (z) and considering the functions which map onto the conic domain k,δ such that ∈ k,δ

Summary

The functions f

Respectively, take all the values in the conic domain k,δ defined by k,δ = u + iv : u > k (u – ) + v + δ with p(z). E onto the conic domain k,δ such that ∈ k,δ. We introduce a function (z pF (a, b, c; z))– given by zp F (a, b, c; z) zp F (a, b, c; z). This operator was recently introduced in [ ]. For p = , this operator is studied by Noor [ ]. For a = n + p, b = c = , this operator was investigated by Liu [ ] and Liu and Noor [ ]

Simple computations yield

Lemma for

This implies that