A subclass of analytic functions defined by the Dziok-Raina operator
© Al-dweby and Darus; licensee Springer 2013
Received: 16 November 2012
Accepted: 5 April 2013
Published: 19 April 2013
The main object of the present paper is to introduce a subclass of analytic functions using the Dziok-Raina operator associated with the quasi hypergeometric functions. This class generalizes some well-known classes of starlike and convex functions. The integral means inequalities and the -neighborhood of this class are considered. Further, some results concerning the n th-Cesaro means of quasi hypergeometric functions for the class above mentioned are considered.
Keywordsquasi hypergeometric functions Dziok-Raina operator Cesaro means integral means
Quasi hypergeometric functions are known as Fox-Wright functions and they appeared as an extension of a generalized hypergeometric functions. Recently, these functions have been given considerable attention by theoretical physicists. Indeed, those functions play an important role in conformal field theory and fractional exclusion statistics such as the quasi-algebraic functions and the partition functions. For a mathematical background for these functions, see [1, 3].
We recall the Dziok-Raina linear operator  as follows:
It should be remarked that the linear operator (1.3) is a generalization of many operators considered earlier. For () and (), , we obtain the Dziok-Srivastava linear operator . This includes (as its special cases) various other linear operators, for example, the ones introduced and studied by Ruscheweyh , Carlson-Shaffer  and Bernardi-Livingston operators [8–10]. Also, many interesting subclasses of analytic functions associated with the operator (1.3) and one may refer to [11, 12].
Lemma 1.1 
Now using , we define the following subclass of analytic functions.
a subclass of being introduced and studied by Silverman .
By suitable choices of the values r, s, , , , , Φ, Ψ, and α, we obtain various subclasses. As illustrations, we present some examples.
where and is the subclasses of that are starlike of order δ and convex of order δ, respectively, which were studied by Silverman .
where , .
Proof The above condition is necessary and sufficient for f to be in the class . To prove this theorem, we use similar arguments as given by Darus . □
Then Theorem 1.2 can be obtained from Theorem 4 in .
2 Integral means inequalities
In the following theorem, we obtain the integral means inequality for the class . We first state a lemma given by Littlewood  as follows.
The next theorem is as the following.
By the definition of subordination, we have (2.3) and this completes the proof. □
In the view of last theorem, we state the next corollaries.
Then for , , (2.4) holds true.
Then for , , (2.4) holds true.
Then Theorem 2.2 can be obtained from Theorem 7 in .
3 Neighborhoods of the class
Now, we investigate -neighborhood for functions in the class .
and is defined as in (2.1).
This gives that . □
4 Cesaro means
In this section, we investigate results on Cesaro means for the function ψ defined by the form (1.2) and for the class . Quasi hypergeometric functions were considered as a generalization to the generalized hypergeometric functions studied by Ruscheweyh , which he observed the following results.
Lemma 4.1 Let if or then the function of the form , is convex.
From the above definitions it is easily to observe the following lemma.
Lemma 4.3 
If , and then .
If , and (the class of Caratheodory functions) with , then , where closed convex hull of .
where k is a positive number and .
Clearly that is the n th Cesaro mean of the geometric series of order β.
Lemma 4.4 
Now, we consider the n th Cesaro means for functions in the class .
Theorem 4.6 If , then the series .
We have studied a class of analytic functions defined by means of the familiar quasi hypergemetric functions. The necessary and sufficient conditions for a function to be in the class are obtained. Several properties for functions belonging to this class are derived. Cesaro results are also being considered. Few other results related to Cesaro means can be seen in [27–31].
The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02 and GUP-2012-023.
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