Abstract
In light of a certain sort of fractional calculus, a generalized symmetric fractional differential operator based on Raina’s function is built. The 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 differential operator. The 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 coefficient differential operator is also created. The geometric and analytic properties of the object are being addressed. The symmetric differential operator in a complex domain is shown to be a generalized fractional differential operator. Finally, we explore the characteristics of the Raina’s symmetric differential operator.
1. 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 fields 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 defined by using some special functions, which are satisfying the symmetric behavior. The Mittag–Leffler function and its extensions, including Raina’s functions, are solutions for all categories of fractional differential equations (see [2–8]).
We examine how Raina’s function may utilize to expand a symmetric fractional differential operator in a complex domain in this research. A range of new normalized analytic functions are explained using the fractional symmetric operator. The idea of differential subordination and superordination is applied to study a collection of differential inequalities. The 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.
2. Methods and Techniques
We will go through the strategies we used in this part.
2.1. Geometric Concepts
We start by the following definition [9]: Concept 2.1. The analytic functions in are subordinated or If for an analytic function owning Concept 2.2. Consider the subclass of analytic functions by satisfying . Furthermore, the functions are called convoluted if they admin the operation [10] Concept 2.3. The class of star-like functions and the class of convex univalent functions are both related to the class of normalized analytic functions . In addition, we require the class of analytic functions
2.2. Modified Special Function
Special functions include integrals and the outputs of many different types of differential equations. Therefore, 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. Concept 2.4. The extended Mittag–Leffler function is formulated by the series [12] such that . Clearly, for , implies that After that, we will go through Raina’s function. Concept 2.5. Raina’s function is determined by the power series as follows [12]: such that and is a collection of real or complex numbers. Notice 2.6. We have the following well-known special cases:(i)(ii)(iii)
Employing the functional , we get the convolution operator, for such that
Clearly, . From the above structure, the fractional differential operator can be viewed geometrically.
Note that the operator is a new type of the convoluted Carlson–Shaffer operator [13] satisfying , and , with
Moreover, when for all , we have the Sàlàgean operator [14]:
2.3. Arguments
The following precursors are utilized to develop the results of this inquiry into differential subordination theory: Argument 2.7. (see [9]). Suppose that and are convex univalent in with . Then, for a fixed value the subordination gives Argument 2.8. (see [9]). Consider the class of holomorphic functions as follows: where and . The condition implies In addition, if and , then for satisfying Argument 2.9 (see [15]). Let , where and for . Then, the subordination yields Argument 2.10. (see [16]). Let , where and the functional is univalent for some positive fixed number . Then the differential inequality implies
3. Consequences
The next class of normalized analytic functions is defined in this paper, and its features are investigated employing differential subordination and superordination theory. Concept 3.1. A function aims to be in the class if it fulfills the inequality whenever .
Eventually, the convexity of the univalent functionimplies that
Consider the functional , 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.
3.1. Results of Subordination Formula
We begin with the following outcome.
Proposition 1. Assume that . Ifthen the upper bound of the coefficients is determined by the probability measure :
In addition, ifthen and
Proof. Suppose thatContinuously, the Carathéodory positivist lemma entailswhere is a probability measure. Besides, ifthen according to Theorem 1.6 in [10] and for a fixed number , we haveHence, .
The following findings reveal the functional sandwich theory’s required and adequate methodology.
Proposition 2. Suppose thatwhere and convex in . Moreover, let be univalent in with , where indicates the class of all univalent analytic functions having the limit and
Thenand is the best subdominant and is the best dominant.
Proof. Puttinga calculation yieldsAs a consequence, the double inequality produced is as follows:Finally, Arguments 2.9 and 2.10 provide the required outcome. □
Proposition 3. Assume thatThen this leads to
Proof. A calculation gives thatIn view of Argument 2.8, we obtain
3.2. 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 to extend GKSs. We deal with the upper bound solution, for the fractional differential equation
The outcome of (45) is formulated as follows.
Proposition 4. Let . Then (45) has a solution expressed bywhere is a constant and is the hypergeometric function.
Proof. Let . Thus, we obtainwhere and . As a result, we get the integral formulaIn view of Schwarz lemma, we get (see Theorem 5.34 in [17]). Therefore, by assuming , we obtain the differential equationIf we reorganize the previous equation, we conclude thatThen multiplying by the functionalwe obtainAs a result, we receive the solution
Example 1. Let , where and . According to Proposition 5, we haveLet . ThenComparing the right sides of the above equations, we obtain that . But is the optimal convex function in the open unit disk; thus, the operator is convex whenever is convex (see Figure 1).
(a)
(b)
(c)
(d)
(e)
(f)
3.3. Symmetric Differential Operator
The Raina’s convoluted operator is assumed to present an extended symmetric differential operator.
When , we have the symmetric operator in [18]. Moreover, when and , we receive the Sàlàgean integral operator [14].
The following classes will be studied: Concept. Let . Then, we define the subclass of star-like functions:(i) if and only if there occurs a convex function satisfying the subordination(ii) if and only if
Proposition 5. Consider . Thenwhere is analytic in with and . Additionally, for , satisfies the inequality
Proof. Because , then we conclude thatThis leads to the existence of a Schwarz function with and such thatwhich implies thatIntegration implies thatA computation bringsThe subordination yieldsMoreover, the disk is mapped by . When we apply to an area that is convex and symmetric with respect to the real axis, we getwhich bringsEmploying equation (65), we obtainAs a result, we get the inequalityHence, we receive
Proposition 6. Suppose that then the odd functionfulfills the inequality
Proof. By the condition , we obtain the existence of a function such thatThis leads toAlso, becausewhere is univalent, then the above subordination yieldsAdditionally, the function is star-like in , which gives the inequalityAs a consequence, we confirm the existence of Schwarz function such thatwhich yields that there is such thatBy rearranging the above inequality, we receiveHence, we have the following conclusion:orThis yieldsAs a result, we obtain the next outcomes.
Corollary 1 (see [18]).. Let in Proposition 6. Then
Corollary 2 (see [19]).. Let and in Theorem 3.9. Then
Corollary 3 (see [20]).. Let and in Theorem 3.9. Then
Corollary 4. Let and in Theorem 3.9. Then
4. Conclusion
The preceding study used symmetric derivative and Jackson’s calculus to generalize Raina’s transformations in . We used the suggested linear convolution operator on the normalized subclass. The operator is utilized to analyze the outcome of a specific form of GKS, which is utilized as an application. The hypergeometric function was used to determine the behavior of solutions. We further stressed that the answer belongs to the normalized analytic functions category.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly to writing this article. All authors read and agreed to the published version of the manuscript.