More General Weighted-Type Fractional Integral Inequalities via Chebyshev Functionals

: The purpose of this research paper is ﬁrst to propose the generalized weighted-type fractional integrals. Then, we investigate some novel inequalities for a class of differentiable functions related to Chebyshev’s functionals by utilizing the proposed modiﬁed weighted-type fractional integral incorporating another function in the kernel F ( θ ) . For the weighted and extended Chebyshev’s functionals, we also propose weighted fractional integral inequalities. With speciﬁc choices of (cid:118) ( θ ) and F ( θ ) as stated in the literature, one may easily study certain new inequalities involving all other types of weighted fractional integrals related to Chebyshev’s functionals. Furthermore, the inequalities for all other type of fractional integrals associated with Chebyshev’s functionals with certain choices of (cid:118) ( θ ) and F ( θ ) are covered from the obtained generalized weighted-type fractional integral inequalities.


Introduction
In [1], for two integrable functions Z 1 and Z 2 on [v 1 , v 2 ], the Chebyshev functional and the weighted Chebyshev functional are respectively proposed as: (1) and: (2) where the functionh 1 is positive and integrable on [v 1 , v 2 ]. In the study of probability and statistical problems, (2) has several applications. In addition, the functional (2) has applications in the domain of integral and differential equations. Readers may refer to [2][3][4].
In [9], the authors established the following fractional integral inequality for the Chebyshev functional (2) by: [15,16], the extended Chebyshev functional was presented as: This paper is organized as follows: The generalized weighted-type fractional integral inequalities connected to the functionals (1) and (2) are discussed in Section 2. We propose some generalized weighted-type fractional integral inequalities connected to (3) and (4) in Section 3. Finally, in Section 4, we give the concluding remarks.
Here, we define the following generalized weighted-type fractional integral operators.

Generalized Weighted-Type Fractional Integral Inequalities via Chebyshev's Functional
Here, we develop weighted-type generalized fractional integral inequalities via Chebyshev's functional.
where Π(1) is defined by: Proof. Define: By Theorem 2, Z 1 and Z 2 fulfil the hypothesis; therefore, we have: Taking the product on both sides of (20) by −1 (θ) and, then, taking the integration of both sides with respect to over (v 1 , θ) and: Again, taking the product of (21) by −1 (θ) taking integration with respect to ζ over (v 1 , θ) and using (9), we obtain: From (16), it becomes: Consequently, it can be written as: According to (22) and (24), we obtain the desired proof.
Setting Theorem 2 for = 1, we obtain the following new result.

Generalized Weighted-Type Integral Inequalities Associated with Weighted and Extended Chebyshev Functionals
In this section, we construct certain weighted-type generalized fractional integral inequalities.
q + 1 q = 1 and 1 r + 1 r = 1, then the following weighted fractional integral inequality holds: Proof. Let us define Conducting the product of (26) by −1 (θ) tegrating with respect to over (v 1 , θ) and using (9), we obtain: Again, taking the product of (27) by −1 (θ) integrating with respect to ζ over (v 1 , θ) and using (9), we have: On the other side, we also have: By employing the Hölder inequality, we have: and: Thus, H can be estimated as: Hence, from (28) and (32), it follows that: By employing the Hölder inequality for the double integral for (33), we obtain: Now, utilizing the following relations: then (34) becomes, From (36), we have: which completes the required result.
If we consider (θ) = 1 in Theorem 3, the following new result can be obtained.

Concluding Remarks
By utilizing the proposed weighted-type generalized fractional integral operator, we established a class of new integral inequalities for differentiable functions related to Chebyshev's, weighted Chebyshev's, and extended Chebyshev's functionals. The obtained inequalities are in more general form than the existing inequalities, which have been published earlier in the literature. Our result's exceptional cases can be found in [5,11,12,[27][28][29][30]. Furthermore, for other types of operators addressed in Remarks 1 and 2, certain new integral inequalities connected to Chebyshev's functional and its extensions given in the literature can be easily obtained. One may investigate certain other types of integral inequalities by employing the proposed operators in the near future.