Convoluted fractional differentials of various forms utilizing the generalized Raina's function description with applications

A generalized differential operator utilizing Raina's function is constructed in light of a certain type of fractional calculus. We next use the generalized operators to build a formula for analytic functions of type normalized. Our method is based on the concepts of subordination and superordination. As an application, a class of differential equations involving the suggested operator is studied. As seen, the solution is provided by a certain hypergeometric function. We also create a fractional coefficient differential operator. Its geometric and analytic features are discussed. Finally, we use the Jackson's calculus to expand the Raina's differential operator and investigate its properties in relation to geometric function theory.


Introduction
Fractional calculus has grown in popularity in recent years, thanks to its applications in science and engineering. Fifty-first-order differential equations are used to model almost all nonlinear physical processes. In terms of the Mittag-Leffler function and its extensions, all classes of fractional differential equations have solutions in terms of this function (the Queen Function of Fractional Calculus) (see [1][2][3]).
Basic power sums and polynomials, particularly the Mittag-Leffler function and its generalizations (Raina's function), as well as polynomials and their implications, are recognized to have extensive applications in various areas of number theory, such as the theory of partitions. Vector calculus, statistical studies, particle physics, optics, fluid studies, mechanical studies, quantum theory and applications, thermal study, and measurements all benefit from these functions (see [4][5][6][7][8][9][10][11][12]). This function has been investigated in different types of inequalities and convex inequalities. Shu-Bo Chen et al. [13] presented an integral formula inequality containing the Raina's function. Chu et al. [14] generalized harmonically ψ-convex with respect to Raina's function on fractal set. Rashid et al. [15] extended the Mittag-Leffler kernel. Mohammed et al. [16][17][18] introduced various studies on the generalized Mittag-Leffler kernel.
In this study, we look at how Raina's function A μ α,β (ζ ) = ∞ n=0 μ(n) (αn + β) ζ n may be used to extend a differential operator in the open unit disk. The fractional differential operator is employed to explain a variety of innovative normalized analytic functions. Therefore, we utilize the convolution product between the normalized Raina's function and analytic function satisfying the normalization equality χ(0) = χ (0) − 1 = 0. To investigate a collection of differential inequalities, the concept of differential subordination and superordination is employed. Furthermore, we investigate the geometric behaviour of the diffusive wave differential equation, a family of analytic functions. The novel convolution linear operator is used for a variety of applications.

Approaches
In this section, we'll go through the approaches we employed.

Raina's function
Integrals and outcomes of many kinds of differential equations fall within the category of special functions. As a result, most integral sets contain descriptions of special functions, and these special functions entail the most fundamental integrals; at the very least, the integral representation of special functions. Because symmetries of differential equations are significant in both physics and mathematics, the theory of special functions is closely connected to several mathematical physics issues [21]. To begin, we'll look at the Mittag-Leffler function, which is a well-known special function.

Definition 2.4:
The power of the generalized Mittag-Leffler function is as follows: is the gamma function and is the Pochhammer operator. Obviously, we have [10] Continue by defining Raina's function.
n a n ζ n , where n := Now by using the Sàlàgean derivative [23], we have n n a n ζ n . . . n k n a n ζ n .

Preparatory
The conclusions of this investigation into the differential subordination theory are established using the following preliminaries:

Outcomes
In this study, we formulate the next class of normalized analytic functions and study its properties in view of the geometric function theory.

Definition 3.1: A function χ ∈ is called to be in the class
where ρ is convex univalent in K.
Obviously, the convex univalent function is a member in the class Consider the functional χ : K → K, as in the following structure: Based on Definition 3.1, we have the following inequality: Our study is as follows:

Inequalities outcomes
We start with the next property of Raina' s operator.
Then the inequality is fully filled by the coefficient boundaries of χ with the probability measure dω(θ): Bζ +1 ) and

Proof:
Since then χ (ζ ) is a Carathéodory function in the open unit disk. Continuously, the Carathéodory positivist methodology brings that where dω is a probability measure. Additionally, if then in virtue of [20, Theorem 1.6] and for fixed number υ ∈ R, we get Moreover, we have from the proof of [20, Theorem 1.6] and that such that the range ( χ * ρ)(K) i s contained in the interior of ρ(K). This yields 0 ∈ ( χ * ρ)(K). Hence, The next outcomes indicate the necessary and sufficient method for the functional sandwich theory.
where O presents the set of all univalent analytic functions g with lim ζ ∈∂O g = ∞ and and p 1 (ζ ) is the best sub-dominant and p 2 (ζ ) is the best dominant.

Proof: Let
A computation implies As a result, the following double inequality is obtained As a conclusion, the desired result is yielded by Lemmas 2.10 and 2.11.

Theorem 3.4: Let
Proof: A computation yields According to Lemma 2.9 with "= 1, we get

Fractional differential equation
In this part, we continue our study using the convolution linear operator. We formulate the operator to present a generalized formula of the diffusive wave differential equation. When inertial acceleration is substantially lower than all other sources of acceleration, or when there is mostly sub-critical flow with low Froude values, the diffusive wave is viable. In light of the suggested operator, we utilize the class μ,k α,β (ς, 1+ζ 1−ζ ) to develop a class of fractional diffusive wave differential equations. We look at the upper bound of the diffusive wave equation. The formula is as follows: The solution to (5) is given by the following result.  1 (a, b, c; ζ ) indicates the hypergeometric function.

First order differential operator
In the next study, we employ the Raina's operator to define a new generalized differential operator. where n n k a n ζ n n n k+1 a n ζ n ] n k n a n ζ n := ζ + ∞ n=2 κ n n k n a n ζ n where κ n := k 1 (ν, ζ ) + nk 0 (ν, ζ ) k 1 (ν, ζ ) + k 0 (ν, ζ ) ; and the functions k 1 , k 0 : [0, 1] × K → K are analytic in K with

It is clear that, for constant coefficients,
Clearly, if λ assumes only non-negative integer values, that is if λ − [[λ]] = 0, α = β = 1, μ(n) = n!, ∀n ≥ 1, then we have the Sàlàgean differential operator [23]. We also have the differential operator in [29], which is based on the same assumptions. In this section, we examine the geometric properties of the complex conformable derivative (7) when applied to functions with a positive real portion.
, ζ ) and by Definition 3.7, we get Obviously, we obtain ⎛ Accordingly, if and only if The convexity of a function is obtained by combining the inequality 8 with the idea of convex functions: But all convex functions are starlike, then we obtain that The inequality 9 occurs if and only if ⎛ and this ends the proof.
The main condition to put on the operator is computed by our second theorem, for the functional to be of positive real part. Theorem 3.9: For a positive number ε ∈ (0, 1) and λ ∈ [0, ∞) let .
Proof: Applying the differential operator rule to .

The convexity of
This ends the proof.

Quantum starlike methodology
Quantum calculus (QC) is a novel field of mathematical analysis and its applications, with applications in physics and mathematics. Jackson [30,31] originally defined and enhanced the functions of q-differentiation and q-integration. The geometric function theory idea of q-calculus was later incorporated by Ismail et al. [32]. QC is now being used by researchers to propose and build new Ma and Minda classes. Seoudy and Aouf [33] suggested a quantum starlike function subclass based on q-derivatives. Recently, Zainab et al. [34] employed a novel curve to create appropriate q-stralikeness criteria. Different types of q-stralik functions dominated by a 2D-Julia set were explored by Samir et al. [35]. Furthermore, QC is used to generalize a variety of differential and integral operators [36][37][38][39][40][41][42].
Definition 3.10: The Jackson derivative may be shown using the difference operator below.
, q ∈ (0, 1) (11) such that The total of the numbers is also included in the Maclaurin's series representation. where Note that where is a constant. Then there's the multiplication rule, which is formulated by multiplying two numbers together We then use the q-parametric Mandelbrot function to formulate our q-starlike class, linking it to the normalized function subclass in the process K We aim to investigate the sufficient conditions on the two parameters and q to obtain the q-starlike function.
The following examples involve the Raina' s operator.

Conclusion
Raina's transformations in K were generalized utilizing conformable calculus and Jackson calculus in the above investigation. The Raina's convolution operator is acted on the normalized subclass. As an application, we considered the proposed linear convolution operator in fractional differential equation, type wave equation. The solution of a certain type of diffusion differential equation, which is utilized as a case study, is determined by the hypergeometric function. More investigation is presented by formulating the Raina's convolution operator in a conformable fractional calculus. We studied the main sufficient conditions to get stralike geometry of the operator (see Theorems 3.8 and 3.9).
Finally, the quantum calculus is utilized to recognize the q-starlike function together with the q-parametric Mandelbort function. As an application, we applied the result using the Raina's convolution operator (see Example 3.12).

Disclosure statement
No potential conflict of interest was reported by the author(s).