On New Modifications Governed by Quantum Hahn’s Integral Operator Pertaining to Fractional Calculus

<jats:p>In the article, we present several generalizations for the generalized Čebyšev type inequality in the frame of quantum fractional Hahn’s integral operator by using the quantum shift operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mtext> </mml:mtext></mml:mrow><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>Ψ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="fraktur">q</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>ς</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi mathvariant="fraktur">q</mml:mi><mml:mi>ς</mml:mi><mml:mo>+</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi mathvariant="fraktur">q</mml:mi></mml:mrow></mml:mfenced><mml:mi>σ</mml:mi></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>ς</mml:mi><mml:mo>∈</mml:mo><mml:mfenced open="[" close="]"><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>ω</mml:mi><mml:mo>/</mml:mo><mml:mfenced open="(" close=")"><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi mathvariant="fraktur">q</mml:mi></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi mathvariant="fraktur">q</mml:mi><mml:mo><</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>ω</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mfenced></mml:math>. As applications, we provide some associated variants to illustrate the efficiency of quantum Hahn’s integral operator and compare our obtained results and proposed technique with the previously known results and existing technique. Our ideas and approaches may lead to new directions in fractional quantum calculus theory.</jats:p>


Introduction
In 1882, Čebyšev discovered a fascinating and significantly valuable integral inequality as follows: if Q and U are two integrable and synchronous functions on ½l 1 , l 2 , where the functions Q and U are said to be synchronous on ½l 1 , for all x, y ∈ ½l 1 , l 2 .
Quantum difference operators are receiving an increase of interest due to their applications [21,22]. Roughly speaking, quantum calculus can substitute the classical derivative by a difference operator, which allows to deal nondifferentiation functions.
The Hahn difference operator (3) unifies (in the limit) the Jackson q-difference derivative D q [24] for q ∈ ð0, 1Þ and the forward difference D ω for q ⟶ 1, which are defined by if H 1 ′ð0Þ exists for ω = 0, and for ω > 0. The Hahn difference operator has been applied successfully in the construction of families of orthogonal polynomials as well as in approximation problems [25][26][27][28].
In [29], the authors introduced some concepts of fractional quantum calculus in terms of a q-shifting operator σ Ψ q ðςÞ = qς + ð1 − qÞσ.
Let I = ½l 1 , l 2 ⊆ ℝ be an interval. Then, the point σ of Hahn calculus on the interval ½l 1 , l 2 generated by the quantum numbers 0 < q < 1 and ω ≥ 0 is given by We state that σ ∈ ½l 1 , l 2 for all consequences of our investigation; the quantum Hahn shifting operator is defined by and the iterated κ-times quantum shifting is given by with σ Ψ 0 q ðςÞ = ς for ς ∈ ½l 1 , l 2 . Let us recall the basic knowledge of quantum Hahn calculus on an interval ½l 1 , l 2 (see [30]). Definition 1. Let H 1 be a function defined on ½l 1 , l 2 . Then, the quantum Hahn difference operator is defined by if H 1 is differentiable at σ. Definition 2. Let H 1 : ½l 1 , l 2 ⟶ ℝ be a given function and x, y ∈ ½l 1 , l 2 . Then, the q, ω-quantum Hahn integral of H 1 from x to y is defined by for ς ∈ ½l 1 , l 2 provided that the series converge at ς = x and ς = y. The function H 1 is said to be q, ω-integrable on ½l 1 , l 2 if (11) exists for all ς ∈ ½l 1 , l 2 .
Definition 3 (see [31]). Suppose that α ≥ 0 and H 1 : ½l 1 , l 2 ⟶ ℝ is a real-valued function. Then, the fractional quantum Hahn derivative ð l 1 D α q,ω H 1 ÞðςÞ of the Riemann-Liouville type of order α is defined by where n is the smallest integer greater than or equal to α.
Asawasamrit et al. [76] expounded the concept of qderivative over the interval ½l 1 , l 2 ⊂ ℝ and derived several inequalities on quantum analogues, for example, q-Cauchy-Schwarz inequality, q-Grüss-Čebyšev integral inequality, q-Grüss inequality, and other integral inequalities, by use of the convexity theory.
The main purpose of the article is to provide the novel versions of the generalized Čebyšev inequalities and present the associated variants via quantum Hahn's fractional integral operator.
To end this section, we give the definition of the onesided fractional quantum Hahn integral in the Riemann-Liouville sense. Definition 6. Let α ≥ 0 and H 1 : ½l 1 , l 2 ⟶ ℝ be a real-valued function. Then, the one-sided fractional quantum Hahn integral ð 0 + I α q,ω H 1 ÞðςÞ of Riemann-Liouville type of order α is defined by

Certain Extended Weighted Čebyšev Fractional Quantum Hahn Integral Operator
In this section, we provide several new generalizations for the weighted extensions of Čebyšev functionals via a quantum Hahn integral operator.

Journal of Function Spaces
Multiplying both sides of (21) by ððς− σ Ψ q ðxÞÞ α−1 σ /Γ q ðαÞÞ H 1 ðxÞ and then performing the q, ω-integration with respect to x over ð0, ςÞ, we have Inequality (22) can be rewritten as Multiplying both sides of (23) by ððς− σ Ψ q ðyÞÞ α−1 σ /Γ q ðαÞÞ H 1 ðyÞ and then performing the q, ω-integration with respect to y over ð0, ςÞ, we get Similarly, we have Taking into account the Hölder inequality, we have It follows from (26) and (27) that From (24) and (28), we obtain Making use of the Hölder inequality for bivariate integral, we have It follows from (30) and the inequalities Journal of Function Spaces Therefore, we get the desired inequality Let ω = 1. Then, Theorem 7 leads to Corollary 8 which provide a new result for q-fractional integral operator.hold for all ς > 0: 1Þ, H 1 be a positive q, 1integrable function defined on ½0, ∞Þ, and Q and U be two q, 1-differentiable functions defined on ½0, ∞Þ such that Q ′ ∈ L s ð½0, ∞ÞÞ and U ′ ∈ L r ð½0, ∞ÞÞ. Then, the inequalities hold for all ς > 0.
Remark 9. If q = ω = 1, then Theorem 7 reduces to the result for the Riemann-Liouville fractional integral operator given in [77]. Some results given in the literature [13,78] can also be obtained from Theorem 7 immediately.

Journal of Function Spaces
Let ω i = 1 for i = 1, 2. Then, Theorem 10 leads to Corollary 11 which provide a new result for q i -fractional integral operator.
Proof. From (49) and (50), we clearly see that Conducting product on both sides of (51) by ðς − σ Ψ q ðxÞÞ ðα−1Þ σ /Γ q ðαÞ and then performing the q, ω-integration with respect to x over ð0, ςÞ yield It follows from eE > 0 and From (52) and (54), we conclude that which implies (A 9 ). By making few changes in (A 9 ), we can get (A 10 ) and (A 11 ).
Theorem 16. Let q ∈ ð0, 1Þ, ω ≥ 0, and Q and U be the positive q, ω-integrable functions defined on ½0, ∞Þ such that Then, the inequalities  Proof. It follows from (56) and (57) that The proof can be derived by following Theorem 15.
Replacing, respectively, P and Q by P QU and Q/U in (61) and (56), we attain the required inequality (A 16 ): Theorem 18. Let q ∈ ð0, 1Þ, ω ≥ 0, and P , Q and U be the q, ω-integrable functions defined on ½0, ∞Þ such that P ðxÞ ≥ 0 and (56) holds. Then, for all ς > 0, we have Proof. We clearly see that Inequality (63) can be written as Conducting product on both sides of (64) by ðς − σ Ψ q ðxÞÞ ðα−1Þ σ /Γ q ðαÞ and then performing the q, ω-integration with respect to x over ð0, ςÞ yield Also, by Cauchy inequality, we get Multiplying both sides of the inequality (66) by 1/ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , we get ðA 17 Þ.

Conclusion
We have discovered several generalizations for the generalized Čebyšev type inequality via quantum fractional Hahn's integral operator by using the quantum shift operator σ Ψ q ðςÞ = qς + ð1 − qÞσðς ∈ ½l 1 , l 2 , σ = l 1 + ðω/ð1 − qÞÞ, 0 < q < 1, ω ≥ 0Þ, provided some associated variants to show the efficiency of quantum Hahn's integral operator, and compared our obtained results and proposed technique with the previously known results and existing technique. The outcome shows that the proposed plans are extremely important and computationally appealing to deal with several sorts of differential equations. As a future research course of this paper, the new techniques obtained in the present paper can be prolonged to attain analytical solutions of quantum mechanics introduced in different works distributed currently connected with high-dimensional fractional equations.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.