On New Modifications Governed by Quantum Hahn’s Integral Operator Pertaining to Fractional Calculus
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 . 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.
In 1882, Čebyšev discovered a fascinating and significantly valuable integral inequality as follows: if and are two integrable and synchronous functions on , where the functions and are said to be synchronous on if for all .
It is well known that the Čebyšev inequality (1) has wild applications in the fields of pure and applied mathematics [1–10]. Recently, the generalizations and variants for the Čebyšev inequality (1) have attracted the attention of many researchers [11–20].
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.
Let , , be an interval such that , and be a real-valued function. Then, the Hahn difference operator  is defined by if is differentiable at .
In , the authors introduced some concepts of fractional quantum calculus in terms of a -shifting operator .
Let be an interval. Then, the point of Hahn calculus on the interval generated by the quantum numbers and is given by
We state that for all consequences of our investigation; the quantum Hahn shifting operator is defined by and the iterated -times quantum shifting is given by with for .
Let us recall the basic knowledge of quantum Hahn calculus on an interval (see ).
Definition 1. Let be a function defined on . Then, the quantum Hahn difference operator is defined by if is differentiable at .
Definition 2. Let be a given function and . Then, the -quantum Hahn integral of from to is defined by where for provided that the series converge at and . The function is said to be -integrable on if (11) exists for all .
Before approaching the main definitions of fractional quantum Hahn calculus on , we present the -power function which is stated as
Precisely, if , then with for .
The -gamma function is defined as for .
Obviously, , where and is the quantum number.
Now, we introduce the concepts of fractional quantum Hahn derivative and integral of Riemann-Liouville type .
Definition 3 (see ). Suppose that and is a real-valued function. Then, the fractional quantum Hahn derivative of the Riemann-Liouville type of order is defined by where is the smallest integer greater than or equal to .
Definition 4 (see ). Let and be a real-valued function. Then, the fractional quantum Hahn integral of the Riemann-Liouville type of order is defined by with if the right-hand side exists.
Theorem 5 (see ). Let , , and . Then, one has
Fractional calculus is invariably important in almost all fields of mathematics and applied sciences. Also, the fractional differential equations can provide adequate models for many physical problems in areas such as heat equation, wave equation, Poisson equation and Laplace equation, fluid mechanics, biological populations, viscoelasticity, advection-diffusion, and signal processing [32, 33].
Inequality plays an irreplaceable role in the development of mathematics. Very recently, many new inequalities such as Hermite-Hadamard type inequality [34–38], Petrović type inequality , Pólya-Szegö type inequality , Ostrowski type inequality , reverse Minkowski inequality , Jensen type inequality [43, 44], Bessel function inequality , trigonometric and hyperbolic function inequalities , fractional integral inequality [47–51], complete and generalized elliptic integral inequalities [52–57], generalized convex function inequality [58–60], and mean value inequality [61–63] have been discovered by many researchers. In particular, the applications of integral inequalities have gained considerable importance among researchers for fixed-point theorems; the existence and uniqueness of solutions for differential equations [64–68] and numerous numerical and analytical methods have been recommended for the advancement of integral inequalities [69–75].
Asawasamrit et al.  expounded the concept of -derivative over the interval and derived several inequalities on quantum analogues, for example, -Cauchy-Schwarz inequality, -Grüss-Čebyšev integral inequality, -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 one-sided fractional quantum Hahn integral in the Riemann-Liouville sense.
Definition 6. Let and be a real-valued function. Then, the one-sided fractional quantum Hahn integral of Riemann-Liouville type of order is defined by
2. 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.
Theorem 7. Let with , , , be a positive -integrable function defined on , and and be two -differentiable functions defined on such that and . Then, the inequalities hold for all .
Proof. Let and Then, can be written as Multiplying both sides of (21) by and then performing the -integration with respect to over , we have Inequality (22) can be rewritten as Multiplying both sides of (23) by and then performing the -integration with respect to over , 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 that Therefore, we get the desired inequality
Corollary 8. Let with , , be a positive -integrable function defined on , and and be two -differentiable functions defined on such that and . Then, the inequalities hold for all .
Remark 9. If , then Theorem 7 reduces to the result for the Riemann-Liouville fractional integral operator given in . Some results given in the literature [13, 78] can also be obtained from Theorem 7 immediately.
Theorem 10. Let with , , , and be the positive -integrable functions defined on , and and be the -differentiable functions defined on such that and . Then, the inequality holds for all .
Corollary 11. Let with , , and be the positive -integrable functions defined on , and and be the -differentiable functions defined on such that and . Then, the inequality holds for all .
3. Some New Generalizations by Fractional Quantum Hahn Integral Operator
Theorem 13. Let with , and for , and and be the the positive -integral functions. Then the following inequalities
hold for all .
Proof. Considering the well-known Young inequality
for all and with .
Substituting and into (39) gives Multiplying both sides of (40) by and then performing the -integration with respect to and over lead to the conclusion that