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

Fractional calculus (real and complex) is a rapidly growing subject of interest for physicists and mathematicians. e 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. e classical de�nitions 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 de�ned the fractional operators (derivative and integral) in the complex zz-plane C as follows. �e�nition �� e fractional derivative of order αα is de�ned, for a function fffzzf by


Introduction
Fractional calculus (real and complex) is a rapidly growing subject of interest for physicists and mathematicians. e 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][2][3][4][5][6][7].
Recently, the theory of fractional calculus has found interesting applications in the theory of analytic functions. e classical de�nitions 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 de�ned the fractional operators (derivative and integral) in the complex -plane ℂ as follows.
�e�nition �� e fractional derivative of order is de�ned, for a function by where the function is analytical in simply-connected region of the complex -plane ℂ containing the origin and the multiplicity of − − is removed by requiring log − to be real when − 0.
�e�nition �� e fractional integral of order is de�ned, for a function , by where the function is analytical in simply connected region of the complex -plane ℂ containing the origin and the multiplicity of − −1 is removed by requiring log − to be real when − 0.

International Journal of Analysis
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 ℂ containing the origin and the multiplicity of ( ) is removed by requiring log( ) to be real when ( ) > 0. When 0, 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 ℂ containing the origin and the multiplicity of ( ) is removed by requiring log( ) to be real when ( ) > 0. 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 ℜ( ( ) ( )) > 0, and convex ℜ( ( ( ) ( ))) > 0 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 Ψ [ of the hypergeometric function de�ned by [12] where > 0 for all , > 0 for all , and ∑ ∑ ≥ 0 for suitable values | | , and are complex parameters. It is well known that and is the generalized hypergeometric function. Now by making use of the operator (7), we introduce the following extension operator Φ ∶ : Obviously, when , we have the extension fractional differential operator de�ned 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.
e function is said to be in the class Consequently, from De�nition 4, we have where ( ) ⋯ satis�es the following properties [26]: We denote this class by .

Note that
4 International Journal of Analysis (see [27]), (see [28]). It is well known that, for the univalent function of the form (9), the sharp inequality | 3 − 2 2 | ≤ 1 holds. In the recent paper, we assume the Hankel determinant for = 2 = 2 and calculate the sharp bound for the functional | 2 4 − 2 3 | for ℛ ( ). 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 . en, the Hadamard product
is also univalent convex function in .

Main Results
We have the following result.
). en the integral is also in ( ).
Proof. It is easy to show that erefore, But ( ) ( / ), thus in view of Lemma 8, the proof is complete.
where is de�ned in eorem 16.
Proof. Since ( ), then we have But is univalent convex function (eorem 16); thus by an application of Lemma 7, we obtain the desired assertion.

Conclusion
We de�ned 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 de�ned and studied. e sharp bound for nonlinear functional based on the second-order Hankel determinant | 4 3 |, 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 eorems 11 and 16 based on the subordination concept and employing the result due to Ponnusamy and Singh.