A Class of Symmetric Fractional Differential Operator Formed by Special Functions

In light of a certain sort of fractional calculus, a generalized symmetric fractional dierential operator based on Raina’s function is built. e generalized operator is then used to create a formula for analytic functions of type normalized. We use the ideas of subordination and superordination to show a collection of inequalities using the suggested dierential operator. e new Raina’s operator is also used to the generalized kinematic solutions (GKS). Using the concepts of subordination and superordination, we provide analytic solutions for GKS. As a consequence, a certain hypergeometric function provides the answer. A fractional coecient dierential operator is also created. e geometric and analytic properties of the object are being addressed. e symmetric dierential operator in a complex domain is shown to be a generalized fractional dierential operator. Finally, we explore the characteristics of the Raina’s symmetric dierential operator.


Introduction
Symmetry is both an abstract basis of attractiveness and an applied tool for resolving convoluted problems. As a consequence, symmetry is a well-known foundation in numerous elds of physics. Despite a well-developed abstract theory of analytic symmetry, symmetry in real-world complex networks has established little attention [1]. Many scientists in many domains of mathematical sciences have been interested in learning more about the theory of symmetric operators. A special class of symmetric operators is de ned by using some special functions, which are satisfying the symmetric behavior. e Mittag-Le er function and its extensions, including Raina's functions, are solutions for all categories of fractional di erential equations (see [2][3][4][5][6][7][8]).
We examine how Raina's function may utilize to expand a symmetric fractional di erential operator in a complex domain in this research. A range of new normalized analytic functions are explained using the fractional symmetric operator. e idea of di erential subordination and superordination is applied to study a collection of di erential inequalities. e geometric behavior of the generalized kinematic solution (GKS), a family of analytic solutions, is also studied. A variety of applications employ the new convolution linear operator. (1) If for an analytic function υ, |υ| ≤ |ξ| < 1 owning (2) Concept 2.2. Consider the subclass of analytic functions Λ by satisfying ψ(0) � 0, ψ ′ (0) � 1. Furthermore, the functions ψ 1 , ψ 2 ∈ Λ are called convoluted (ψ 1 * ψ 2 ) if they admin the operation [10] Concept 2.3. e S * class of star-like functions and the C class of convex univalent functions are both related to the class of normalized analytic functions (Λ). In addition, we require the class of analytic functions

Modified Special Function.
Special functions include integrals and the outputs of many different types of differential equations. erefore, most integral sets include special duty descriptions, and these duties include the elementary integrals. Since symmetries are important in real life, the philosophy of special functions is tightly linked to various mathematical physics topics [11]. We will start with a well-known special function, the Mittag-Leffler function.
Argument 2.10 (see [16]). Let φ, ϕ ∈ Π[b, n], where ϕ ∈ C and the functional φ(ξ) + vξφ ′ (ξ) is univalent for some positive fixed number v. en the differential inequality implies 3. Consequences e next class of normalized analytic functions is defined in this paper, and its features are investigated employing differential subordination and superordination theory.
Eventually, the convexity of the univalent function implies that Consider the functional Σ λ ψ : U ⟶ U, as in the following structure: Consequently, in view of Concept 3.1, we get the next inequality We proceed to investigate the geometric possessions of the suggested operators.

Results of Subordination Formula.
We begin with the following outcome.
Proof. Putting a calculation yields As a consequence, the double inequality produced is as follows: Finally, Arguments 2.9 and 2.10 provide the required outcome. □ □ Proposition 3. Assume that en this leads to Proof. A calculation gives that 4 Journal of Mathematics In view of Argument 2.8, we obtain

Fractional Differential Equation with Kinematic
Solutions. We will use the generalized differential operator to continue our research in this section. A generalized formula for the kinematic solutions (GKS) is presented using the suggested operator. Kinematic behaviors describe the motion of an item with constant acceleration in a dynamic system. We aim to utilize the class Ω μ α,β (λ, (1 + ξ/1 − ξ)) to extend GKSs. We deal with the upper bound solution, for the fractional differential equation e outcome of (45) is formulated as follows.

Symmetric Differential Operator.
e Raina's convoluted operator is assumed to present an extended symmetric differential operator.