International Journal of Analysis

Volume 2013, Article ID 542828, 7 pages

http://dx.doi.org/10.1155/2013/542828

## Bounded Nonlinear Functional Derived by the Generalized Srivastava-Owa Fractional Differential Operator

Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia

Received 25 November 2012; Revised 6 January 2013; Accepted 6 January 2013

Academic Editor: Alain Miranville

Copyright © 2013 Rabha W. Ibrahim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By making use of the generalized Srivastava-Owa fractional differential operator, a class of analytical functions is imposed. The sharp bound for the nonlinear functional associated with the Hankel determinant is computed. We consider a new technique to prove our results. Important properties such as inclusion, subordination, and Hadamard product are studied. Some recent results are included.

#### 1. Introduction

Fractional calculus (real and complex) is a rapidly growing subject of interest for physicists and mathematicians. The reason for this is that problems may be discussed in a much more stringent and elegant way than using traditional methods. Fractional differential equations have emerged as a new branch of applied mathematics which has been used for many mathematical models in science and engineering. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. Several different derivatives were introduced: Riemann-Liouville, Hadamard, Grunwald-Letnikov, Riesz, Erdelyi-Kober operators, and Caputo [1–7].

Recently, the theory of fractional calculus has found interesting applications in the theory of analytic functions. The classical definitions of fractional operators and their generalizations have fruitfully been employed for imposing, for example, the characterization properties, coefficient estimates [8], distortion inequalities [9], and convolution structures for various subclasses of analytic functions and the works in the research monographs. In [10], Srivastava and Owa defined the fractional operators (derivative and integral) in the complex -plane as follows.

*Definition 1. *The fractional derivative of order is defined, for a function by
where the function is analytical in simply-connected region of the complex -plane containing the origin and the multiplicity of is removed by requiring to be real when .

*Definition 2. *The fractional integral of order is defined, for a function , by
where the function is analytical in simply connected region of the complex -plane containing the origin and the multiplicity of is removed by requiring to be real when .

In [11], the author generalized a formula for the fractional integral as follows: for natural and real , the -fold integral of the form Employing the Dirichlet technique implies Repeating the above step times yields which imposes the fractional operator type where and are real numbers and the function is analytic in simply connected region of the complex -plane containing the origin and the multiplicity of is removed by requiring to be real when . When , we arrive at the standard Srivastava-Owa fractional integral. Further information can be found in [11].

Corresponding to the fractional integral operator, the fractional differential operator is where the function is analytical in simply connected region of the complex -plane containing the origin and the multiplicity of is removed by requiring to be real when . We have

Let denote the class of functions normalized by Also, let and denote the subclasses of consisting of functions which are, respectively, univalent, starlike , and convex in . It is well known that, if the function given by (9) is in the class , then . Moreover, if the function given by (9) is in the class , then .

In our present investigation, we will also make use of the Fox-Wright generalization of the hypergeometric function defined by [12] where for all , for all , and for suitable values , and are complex parameters.

It is well known that where and is the generalized hypergeometric function.

Now by making use of the operator (7), we introduce the following extension operator : Obviously, when , we have the extension fractional differential operator defined in [13] ([14] for recent work), which contains the Carlson-Shaffer operator. In term of the Fox-Wright generalized function, where , , and is the Hadamard product. Note that where is the Carlson-Shaffer operator. Moreover, operator (14) can be viewed as a linear operator which is essentially analogous to the Dziok-Srivastava operator whenever used instead of the Fox-Wright generalization of the hypergeometric function.

Recently, various results, such as convolution and inclusion properties, distortion theorem, extreme points, and coefficient estimates, are proposed by many authors for the operators due to Srivastava involving the Wright function, generalized hypergeometric function, and Meijer's -functions. These operators are Dziok-Srivastava, Srivastava-Wright, Cho-Kwon-Srivastava operator, Cho-Saigo-Srivastava operator, Jung-Kim-Srivastava, and Srivastava-Owa operators (see [15–24]). Going on in this generalization, we have finally the Erdelyi-Kober operator of fractional integration with three parameters used in [25].

*Definition 3 (subordination principal). *For two functions and analytical in , we say that the function is subordinated to in and write , if there exists a Schwartz function analytical in with , and , such that . In particular, if the function is univalent in , the above subordination is equivalent to and .

*Definition 4. *For the function defined by (9), the Hankel determinant of is defined by
Now we proceed to define a new class of analytic function involving the operator (13).

*Definition 5. *The function is said to be in the class , where , if it satisfies the inequality
Consequently, from Definition 4, we have
where satisfies the following properties [26]:
We denote this class by .

Note that (see [27]),(see [28]).

It is well known that, for the univalent function of the form (9), the sharp inequality holds. In the recent paper, we assume the Hankel determinant for and calculate the sharp bound for the functional for . Properties of this class are illustrated, and some well-known results are generalized. For this purpose, we need the following preliminary in the sequel, which can be found in [29].

Lemma 6. *Let and be univalent convex in . Then, the Hadamard product is also univalent convex function in .*

Lemma 7. *Let and be univalent convex in , and and . Then, .*

Lemma 8. *Let and be starlike of order 12 then, for function satisfying ,
*

#### 2. Main Results

We have the following result.

Theorem 9. *Let the function be in the class . Then
**
where and
**
The estimate (23) is sharp.*

*Proof. *Since , then
Comparing the coefficients of (13) and (25), we receive
where
Therefore, (26) implies
By letting and using (i)–(iii), we have
By employing the triangle inequality and assuming , and , we obtain
Our aim is to maximize in the interior of the domain . Since
thus cannot have a maximum in the interior of . Furthermore,
where
But
hence the upper bound of (28) is
The equality holds for the functions

*Remark 10. *Letting , we receive a recent result due to Mishra and Gochhayat [28]; putting , we obtain a result given by Janteng et al. [30].

Theorem 11. *Assume that and , with . If the subordination
**
where
**
holds, then
*

*Proof. *Let . We rewrite
where
and is a Pochhammer symbol. Therefore, Assumption (37) implies that is convex (see [31, Theorem 1.9]) and consequently (Marx-Strohhäcker Theorem [32]). Moreover, the function is starlike of order 12, then in view of Lemma 8, we obtain that
and consequently .

*Remark 12. *Condition (37) can be replaced by another condition to obtain the convexity of the function , such that
yields that is convex (see [31]).

Theorem 13. *Let and . Then .*

*Proof. *By employing the properties of the Hadamard product, we receive
Therefore,
In virtue of Lemma 8, we have
Hence .

Theorem 14. *Let . Then the integral
**
is also in .*

*Proof. *It is easy to show that
Therefore,
But , thus in view of Lemma 8, the proof is complete.

*Remark 15. *When in Theorems 13 and 14, we have the results given in [28].

Theorem 16. *Let . If the subordination
**
holds, then , is univalent convex function.*

*Proof. *Condition (51) yields that is univalent convex (see [31, Theorem 1.9]) and consequently, in view of Lemma 6, is univalent convex function.

Theorem 17. *Let . Then
**
where is defined in Theorem 16.*

*Proof. *Since , then we have
But is univalent convex function (Theorem 16); thus by an application of Lemma 7, we obtain the desired assertion.

#### 3. Conclusion

We defined a new fractional differential operator which generalized well-known linear and nonlinear operators such as Carlson-Shaffer operator and the Dziok-Srivastava (linear operators) and Srivastava-Owa fractional differential operators (nonlinear operator). By making uses this operator a generalized class of analytic functions is defined and studied. The sharp bound for nonlinear functional based on the second-order Hankel determinant , involving the generalized fractional differential operator, is computed. Several properties, depending on the Hadamard product, are imposed. We have shown that some results are generalized by recent works due to Mishra-Gochhayat, Ling-Ding, and Janteng et al. Furthermore, a new approach is introduced in the proof of Theorems 11 and 16 based on the subordination concept and employing the result due to Ponnusamy and Singh.

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