Certain Quantum Operator Related to Generalized Mittag–Leffler Function

: In this paper, we present a novel class of analytic functions in the form h ( z ) = z p + ∞ ∑ k = p + 1 a k z k in the unit disk. These functions establish a connection between the extended Mittag–Leffler function and the quantum operator presented in this paper, which is denoted by ℵ nq , p ( L , a , b ) and is also an extension of the Raina function that combines with the Jackson derivative. Through the application of differential subordination methods, essential properties like bounds of coefficients and the Fekete–Szeg˝o problem for this class are derived. Additionally, some results of special cases to this study that were previously studied were also highlighted.


Introduction
We shall establish the definition of the class of analytic functions represented by A p as follows where the set D encompasses all the values of z within the open unit disk z ∈ C satisfying |z| < 1.
Given two analytic functions, h 1 and h 2 , within the domain D, the relationship where h 1 is subordinated to h 2 is denoted as h 1 (z) ≺ h 2 (z).This implies the existence of a function ω, known as the Schwarz function, which is analytic in the open disk D and satisfies the criteria ω(0) = 0 and |ω(z)| < 1 for z ∈ D; this function ω further satisfies the condition that h 1 (z) = h 2 (ω(z)) for all z ∈ D. If the function g ∈ S (S is the family of all functions that are univalent in the domain D), then (cf., e.g., [1,2]) see for example [3].
Furthermore, let P := P : P(z) = 1 + b 1 z + b 2 z 2 + . . ., Re P(z) > 0, z ∈ D denote all the Carathéodory functions (see [4,5]).Quantum calculus holds substantial importance in many fields like hypergeometric series theory and quantum physics as well as other physical phenomena.The definition of both q differentiation and q integration was first defined by Jackson ([6,7]).
There are many authors who have studied the operators of quantum calculus through many diverse applications in geometric function theory; e.g., Attiya et al. [8] studied differential operators related to the q-Raina function, Ibrahim [9], Al-shbeil et al. [10] and Karthikeyan et al. [11] studied the q-convolution of a certain class of analytic functions related to the quantum differential operator in GFT, Ismail et al. [12] and Riaz et al. [13] studied starlike functions defined by q-fractional derivatives, Shaba et al. [14] studied coefficient inequalities of q-bi-univalent associated with q-hyperbolic tangent functions, Al-Shaikh et al. [15] studied a class of close-to-convex functions defined by a quantum difference operator, Sitthiwirattham et al. [16] studied Maclaurin's coefficients inequalities for convex functions in q-calculus, Al-Shaikhm [17] studied some classes of analytic functions associated with a Salagean quantum differential operator, and Tang et al. [18] studied the Hankel and Toeplitz determinant for certain subclasses of multivalent q-starlike functions.
We need the following definitions, lemmas and notations to obtain our results in the second and third sections in this paper.Definition 1. Raina's function ( [19]; see also [8]) is defined by using gamma function Γ as follows: where both a and b are complex values in the complex number field C, and provided that Re(a) > 0 and Re(b) > 0, L(k) is a member of the sequence {L(k)} k∈N 0 which is a bounded sequence of arbitrary complex numbers.

We obtain the function E
, in this case, Raina's function simplifies to the Gaussian hypergeometric function described below.
Definition 2 ([6]).The Jackson derivative for the function h(z) is provided as follows: Then, we have In the case where the function h takes the the form (1), then where Also, note that d q κ = 0 and lim where κ represents a constant within the set of complex numbers.
In the case where s ∈ C, the q-shifted factorial is established through the subsequent formula (see [6]): If 1 − q j s , by employing the expression (3), we are able to present the q-shifted gamma function as: For L(0) ̸ = 0, the normalized function L ℵ d,b (see [8]) is defined by If L(k) = (k + 1) −s , s ∈ R, s > 0, a = 0 and b = 1, the operator (4) can be described as the the Sȃlȃgean integral operator with order s (see [30]).
Utilizing the q-gamma function, Attiya et al. [8] introduced the generalized normalized function q,L ℵ d,b (z), which is defined in the following manner: The q-Raina differential operator L ℵ n q : A 1 → A 1 was introduced by Attiya et al. [8] as follows: Therefore, when h belongs to A 1 in the form (1), we have: Now, analogously to L ℵ n q , we introduce a novel operator ℵ n q (L, d, b) for functions in Definition 3. Let the function h(z) ∈ A p be in the form (1). The operator ℵ n q,p (L, d, b) is defined as , in (7), we have the q-differential operator of [31].
(iii) Putting p = 1, q → 1 − and L(k − 1) = 1 in (7), we have a class studied in [33] (see also [34]).Remark 3. Unless otherwise stated in this paper, we will use constraints on the parameters q, n, d, b, L(k) as follows: is a bounded sequence of arbitrary complex numbers with L(0) ̸ = 0.

Definition 4 ([35]
).Let us establish a definition for the convex analytic function γ j,ℑ in domain D as follows: where ℑ ∈ C \ {0}, and the subsequent functions are established by (see [35]) Here, we select ℓ(z where γ j,ℑ in the form (see also [8,35,36]) is given by the Definition 4.
In our paper, we present the class S n,j q,ℑ,p (d, b), which is related to the operator ℵ n q,p (L, d, b).Important properties for the class S n,j q,ℑ,p (d, b) are derived.Also, bounds of coefficients and the Fekete-Szegő problem for this class S n,j q,ℑ,p (d, b) are obtained.Moreover, some results of special cases to this study that were previously studied were also highlighted.

Certain Properties for the
where ω denotes a Schwarz function, z ∈ D. Additionally, if |z| := ϱ < 1, we have By integrating both sides of the equation mentioned above, it can be deduced that then, we obtain Remark 4. Theorem 1 extends the findings of various authors, including the following: 1.
The following theorem and corollaries are related to the coefficient estimation for the class S n,j q,ℑ,p (d, b).
Proof.If we take: p k z k , in the above equation, we will obtain Then, we have By equating the coefficients of z k in the preceding equation, we obtain Consequently, we obtain for some calculation implies that For k = 1, we obtain In the case where k = 2, and employing the aforementioned inequality, then: Assume that for a given value of k ≥ 3, the following inequality holds true through mathematical induction: which completes the proof.
Derived from Theorem 2 as special cases, we yield the following corollaries.

Fekete-Szeg ő Problem Related to the Class S
n,j q,ℑ,p (d,b) In the upcoming theorem, we will provide an estimate for the Fekete-Szegő problem for the class S n,j q,ℑ,p (d, b).
where ω is a Schwarz function.
If v ∈ P is defined by: Therefore, hence, the subsequent coefficients can be established in the following manner: By using some computation, we have where Ψ is defined by (11) and ψ ∈ C. So, by using Lemma 2, we achieve the desired result.
Theorem 3 generalizes some of the previous findings, including the following: where ψ ∈ C, and with ρ 1 and ρ 2 defined by (9).
The following result is related to the sufficient condition of functions in the class S n,j q,ℑ,p (d, b).

Conclusions
Through the utilization of quantum calculus and the generalized Mittag-Leffler function, the operator ℵ n q,p (L, d, b) introduced in Definition 3 is necessary to study the new class S n,j q,ℑ,p (d, b) of analytic functions introduced and investigated in this paper, which is given in Definition 5.By employing the techniques of differential subordination in the geometric function theorem, we derived new and interesting results.In the second section, coefficients inequalities for the class S n,j q,ℑ,p (d, b) and the subordination relation for a special case of this class are obtained.Also, in the third section, the Fekete-Szegő problem and the sufficient conditions for functions in the class S n,j q,ℑ,p (d, b) are derived.Additionally, some results of special cases of the class S n,j q,ℑ,p (d, b) that were previously studied were also highlighted.
If we have two functionsh 1 (z) = z p + ∞ ∑ k=p+1 a k z k and h 2 (z) = z p + ∞ ∑ k=p+1 b k z k of A p ,then the Hadamard product of these functions is defined by:

Definition 6 .
The class S n,j ℑ,p (d, b) is the class S n,j q,ℑ,p (d, b) when q approaches 1 from the left.Lemma 1 ([37]).Consider G(z) = ∞ ∑ k=0 g k z k , which represents a univalent convex function in the domain D, and fulfills the relation:

Theorem 1 .
p (d,b) The theorem presented below gives a new result of functions in the class S n,j ℑ (d, b).If h of the form (1) is in the class S n,j ℑ,p (d, b), then

Theorem 2 .
If h of the form (1) is in the class S n,j q,ℑ,p (d, b), then