On subclasses of analytic functions based on a quantum symmetric conformable differential operator with application

Quantum calculus (the calculus without limit) appeared for the first time in fluid mechanics, noncommutative geometry and combinatorics studies. Recently, it has been included into the field of geometric function theory to extend differential operators, integral operators, and classes of analytic functions, especially the classes that are generated by convolution product (Hadamard product). In this effort, we aim to introduce a quantum symmetric conformable differential operator (Q-SCDO). This operator generalized some well-know differential operators such as Sàlàgean differential operator. By employing the Q-SCDO, we present subclasses of analytic functions to study some of its geometric solutions of q-Painlevé differential equation (type III).


Introduction
The conception of q-calculus model is a creative method for designs of the q-special functions. The procedure of q-calculus improves various kinds of orthogonal polynomials, operators, and special functions, which realize the form of their typical complements. The idea of q-calculus was principally realized by Carmichael [1], Jackson [2], Mason [3], and Trjitzinsky [4]. An analysis of this calculus for the early mechanism was offered by Ismail et al. [5]. Numerous integral and derivative features were formulated by using the convolution concept; for example, the Sàlàgean derivative [6], Al-Oboudi derivative (generalization of the Sàlàgean derivative) [7], and the symmetric Sàlàgean derivative [8]. It is significant to notify that the procedure of convolution finds its uses in different research, analysis, and study of the geometric properties of regular functions (see [9][10][11]). Here, we aim to study some geometric properties of a new quantum symmetric conformable differential operator (Q-SCDO). The classes of analytic functions are suggested by using the convolution product. The consequences are generalized classes in the open unit disk.

Methodology
This section provides the mathematical information that is used in this paper. Let be the category of smooth functions given as follows: where ∪ = {ξ ∈ C : |ξ | < 1}.

Definition 2
For two functions 1 and 2 in , the Hadamard or convolution product is defined as

Definition 4
The q-difference operator of is written by the formula Clearly, we have q ξ n = [n] q ξ n-1 . Consequently, for ∈ , we have For ∈ , the Sàlàgean q-derivative factor [13] is formulated as follows: where k is a positive integer.
A computation based on the definition of q implies that Obviously, the Sàlàgean derivative factor [6].
The value ν = 0 indicates the Sàlàgean derivative Moreover, the following operator can be located in [14], where

Convolution classes
Based on the definition (2.7), we introduce the following classes. Denote the following functions: Thus, in terms of the convolution product, the factor (2.7) is formulated as follows: Let be a function from and σ (ξ ) be a convex univalent function in ∪ such that σ (0) = 1. The class Ξ k q 1 ,q 2 (σ ) is defined by Also, we define a special class involving the above functions when ν → 0, as follows: When k = 0, we have Dziok subclass [15]. We denote by S * (σ ) the class of all functions given by and by C * (σ ) the class of all functions The following preliminary result can be found in [16,17].

Lemma 3.2 For analytic functions
where ξ = re iθ , 0 < r < 1, and p is a positive number.
Some of the few studies in q-calculus are realized by comparison between two different values of calculus. Class Ξ ν,k q 1 ,q 2 (σ ) shows the relation between the q 1 -and q 2 -calculus depending on the operator (2.7).

Inclusions
This section deals with the geometric representations of the class Ξ ν,k q 1 ,q 2 (σ ), q 1 = q 2 and their consequences.
In this place, we note that the conclusion of Theorem 4.1 yields the following consequence: Corollary 4.2 Let be a function from and σ (ξ ) be a convex univalent function in ∪ such that σ (0) = 1. Then In general, we have the following result:
We note that if we replace the condition of Theorem 4.3 by ρ 2 ≺ r ρ 1 such that ρ 1 ∈ C( 1+ξ 1-ξ ) then we obtain the same conclusion.

Integral inequalities
The following section deals with some inequalities containing the operator (2.7). For two functions h(ξ ) = a n ξ n and (ξ ) = b n ξ n , we have h if and only if |a n | ≤ |b n |, ∀n. This inequality is known as the majorization of two analytic functions.
we conclude that [S k ν ] q (ξ ) is majorized by the function σ (ξ , δ) for all δ ≥ 1. By the properties of majorization [18], we have Thus, according to Lemma 3.2, we conclude that In the same manner as in the proof of Theorem 5.1, one can get the next result: If the coefficients of satisfy the inequality | n | ≤ ( 1 nν ) k , ν ∈ (0, 1) then Moreover, the inequality in Theorem 5.1 can be studied in the following result: Consider the operator S κ,k q ψ(z), ψ ∈ Λ. If the coefficients of ψ satisfy the inequality |ϑ n | ≤ ( 1 nκ ) k , κ ∈ (0, ∞) then there is a probability measure μ on (∂U) 2 , for all δ > 1.
Proof Let , ε ∈ ∂∪. Then we have By virtue of Theorem 1.11 in [19], the functional ( 1+ ξ 1+εξ ) δ defines a probability measure μ in (∂∪) 2 fulfilling Then there is a constant c (diffusion constant) such that This completes the proof.

A class of differential equations
This section deals with an application of the operator (2.7) in a class of differential equations (for recent work see [20]). The class of quantum III-Painlevé differential equations has been studied recently in [21][22][23]. This class takes the formula Rearranging Eq. (6.1), we have subjected to the boundary conditions Now by employing the operator (2.7), Eq. (6.2) becomes (called q-Painlevé differential equation of type III) subjected to (6.3). Our aim is to study the geometric solution of (6.4) satisfying the boundary condition (6.3). For this purpose, we define the following analytic class: Definition 6 For a function ∈ and a convex function ψ ∈ ∪ with ψ(0) = 0, the function is said to be in the class V q (ψ) if and only if For the functions in the class V q (ψ), the following result holds.
. Then a straightforward computation implies that Hence, we obtain It is clear that F ∈ H[1, 1] and Then in view of Lemma 6.2, with a = 0, f 1 = 1, f 2 = 0 and f 3 = 1, we have that is, [S k ν ] q ∈ S * with respect to the origin.

Conclusion
In this paper, we presented different types of integral inequalities based on q-calculus and conformable differential operator. These inequalities described the relations between the quantum conformable differential operators for different orders.