A subclass of analytic functions defined by the Dziok-Raina operator
Journal of Inequalities and Applications volume 2013, Article number: 192 (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.
Let denote the class of functions of the form
which are analytic in the open unit disk on the complex plane C. Let , denote the subclasses of consisting of functions, which are starlike of order δ and convex of order δ, respectively. If f and g are analytic in , we say that f is subordinate to g in , written , if and only if there exists the Schwarz function w, analytic in with and in such that (). The convolution (or Hadamard product) of two functions f, g with series expansions and is defined by
A quasi hypergeometric series is a power series in one complex variable z. Let r, s be nonnegative integers and consider the series
where are complex numbers and are positive numbers, which have the relation
This function is a general one in the single variable case. In , the author showed that the above series is convergent for where c denotes the constant
The function satisfies the differential equation
where P defines a fractional derivative operator of order b as the following:
For more details on this operator, see . For and , then the function reduces to the hypergeometric function of higher order
and the above differential equation reduces to ordinary differential equation
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].
Now for , and , let
and ψ is of the form
We recall the Dziok-Raina linear operator  as follows:
For the operator (, ) is defined by the Hadamard product
For a function of the form (1.1) and function ψ of the form (1.2), we derive
where, for convenience,
For the sake of simplicity, we write
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 
For , we have the following:
Now using , we define the following subclass of analytic functions.
Definition 1.1 Given and functions
analytic in such that , , , , we say that is in if and
where is given by (1.3). We further let
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.
Example 1.1 For , , we have
If , then we have the class
Example 1.2 For , , , we obtain
where was studied by Juneja et al. . In particular, for , , , ,
and for , , , , , we have
where and is the subclasses of that are starlike of order δ and convex of order δ, respectively, which were studied by Silverman .
Example 1.3 For , , we get
Example 1.4 For , , , , we obtain
Example 1.5 For , , , , we have
Theorem 1.2 Let a function f be defined by (1.4). Then if and only if
The result is sharp with the extremal functions
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 . □
Remark 1.1 In , the author introduced the class , we observe that if , , and
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.
Lemma 2.1 If the functions f and g are analytic in with , then for and ,
The next theorem is as the following.
Theorem 2.2 Let , be a nondecreasing sequence and be defined by
and is given by
Then for , , we obtain
Proof For a function f of the form (1.4) and , the inequality (2.2) is equivalent to
By Lemma 2.1, it suffices to show that
Setting , we have from (1.5) and (2.3),
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.
Corollary 2.1 Let , and be defined by
Then for , , we obtain
Corollary 2.2 Let and be defined by
Then for , , (2.4) holds true.
Corollary 2.3 Let and be defined by
Then for , , (2.4) holds true.
Remark 2.1 In , the author introduced the class , we observe that if , , and
Then Theorem 2.2 can be obtained from Theorem 7 in .
3 Neighborhoods of the class
For f of the form (1.4), and , Frasin and Darus  investigated the -neighborhood of f as the following:
where p is a fixed positive integer. In particular, for the identity function , we immediately have
Now, we investigate -neighborhood for functions in the class .
Theorem 3.1 If is a nondecreasing sequence, then , where
and is defined as in (2.1).
Proof It follows from (1.5) that if , then
This gives that . □
Corollary 3.1 , where
Corollary 3.2 , where
Corollary 3.3 , where
Corollary 3.4 , where
Corollary 3.5 , where
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.
Note that is the Pochhammer symbol defined by:
Lemma 4.2 Suppose that , then
Now let us recall the following by defining , , and the subclasses of consisting of functions that are starlike in and convex in . By the definitions, we have
We observe that
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 .
Definition 4.1 The n th Cesaro means of order β, of the series of the form (1.2) can be defined as
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 
Let such that for some we have
Theorem 4.5 Let be given in the form (1.2) and convex in and is the nth Cesaro mean of , then
Proof First, we note that , for , and Let be defined such that
In view of Lemma 4.3, the relation (4.2) and the fact that is convex yield that there exists a function and with such that
It is known that and that if and only if . So we have . By using Lemma 4.4, we obtain
Now, we consider the n th Cesaro means for functions in the class .
Let , then the n th Cesaro means of f of order β defined by the form
Theorem 4.6 If , then the series .
and by considering the sufficient condition for f to be in the class we get
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].
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The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02 and GUP-2012-023.
The authors declare that they have no competing interests.
The first author is currently a Ph.D. student under supervision of the second author and jointly worked on deriving the results. All authors read and approved the final manuscript.
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Al-dweby, H., Darus, M. A subclass of analytic functions defined by the Dziok-Raina operator. J Inequal Appl 2013, 192 (2013). https://doi.org/10.1186/1029-242X-2013-192