Univalence of Certain Linear Operators Defined by Hypergeometric Function
© R. Aghalary and A. Ebadian. 2009
Received: 11 January 2009
Accepted: 22 April 2009
Published: 2 June 2009
The main object of the present paper is to investigate univalence and starlikeness of certain integral operators, which are defined here by means of hypergeometric functions. Relevant connections of the results presented here with those obtained in earlier works are also pointed out.
1. Introduction and Preliminaries
For a,b,c ∈ C and c≠ 0,-1,-2,…, the Gussian hypergeometric series F(a,b;c;z) is defined as
For proving our results we need the following lemmas.
Lemma 1.1 (cf. Hallenbeck and Ruscheweyh ).
Lemma 1.2 (cf. Ruscheweyh and Stankiewicz ).
Lemma 1.3 (cf. Ruscheweyh and Sheil-Small ).
2. Main Results
Now the required conclusion follows from (2.13) and (2.14).
Taking in Theorem 2.1 and Corollary 2.2 we get results of .
We follow the method ofproofadopted in .
and the result follows from the last subordination and Corollary 2.2.
It is well-known that (see, ) if and , then is univalent convex function in . So if we take in the Theorem 2.4, we obtain the following.
Taking and on Corollary 2.6, we get a result of .
then we have the following:
In , Pannusamy and Sahoo have also considered the class for the case with
and the result follows from (2.46) and Corollary 2.2.
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