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

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