On integral inequalities related to the weighted and the extended Chebyshev functionals involving different fractional operators

The role of fractional integral operators can be found as one of the best ways to generalize classical inequalities. In this paper, we use different fractional integral operators to produce some inequalities for the weighted and the extended Chebyshev functionals. The results are more general than the available classical results in the literature.


Introduction and preliminaries
Fractional calculus, which is calculus of integrals and derivatives of any arbitrary real or complex order, has gained remarkable popularity and importance during the last four decades or so, due mainly to its demonstrated applications in diverse and widespread fields ranging from natural sciences to social sciences (see, e.g., [1,3,17,19,20,23,24] and the references therein). Beginning with the classical Riemann-Liouville fractional integral and derivative operators, a large number of fractional integral and derivative operators and their generalizations have been presented. Also, many authors have established a variety of inequalities for those fractional integral and derivative operators, some of which have turned out to be useful in analyzing solutions of certain fractional integral and differential equations.
Dahmani et al. [9], established some inequalities for the weighted and the extended Chebyshev functionals with certain conditions via Riemann-Liouville fractional integrals, which are recalled in the following two theorems.

Main results
In this section we present some inequalities for the weighted and the extended Chebyshev functionals involving the fractional integral operators, respectively, Katugampola fractional integral operator, mixed conformable fractional integral operator, and Hadamard fractional integral operator.
Then the left-and right-hand side Katugampola fractional integrals of order (α > 0) of f ∈ X p c (a, b) are defined as follows: with a < x < b and ρ > 0, if the integral exists.  a ρ , b ρ ), r, s, γ > 1 with 1 r + 1 r = 1, 1 s + 1 s = 1, and 1 γ + 1 γ = 1, then for all t > 0, α, ρ > 0, we have and integrating the resulting identity with respect to x from 0 to t, we can write  (t ρ -y ρ ) 1-α p(y) and integrating the resulting identity with respect to y from 0 to t, we can write By Hölder's inequality for double integral, we obtain Using the following properties: (2.9) can be written as This completes the proof.  (a ρ , b ρ ), r, s, γ > 1 with 1 r + 1 r = 1, 1 s + 1 s = 1, and 1 γ + 1 γ = 1, then for all t > 0, α, ρ > 0, we have (t ρ -y ρ ) 1-α q(y) and integrating the resulting identity with respect to y from 0 to t, we can write Using the same arguments as in the proof of Theorem 2.1, we obtain the desired result.

Definition 2.2 ([1]) Let f be defined on [a, b]
and α ∈ C, Re(α) > 0, ρ > 0. Then: and (ii) The mixed right conformable fractional integral of f is defined by For recent results related to this operators, we refer the reader to [1,2,4,26].    Proof Multiplying (2.12) by (log t y ) α-1 y (α) q(y) and integrating the resulting identity with respect to y from 1 to t, we can write Using the same arguments as in the proof of Theorem 2.1, we obtain the desired result.

Concluding remarks
In this paper, we established some integral inequalities related to the weighted and the extended Chebyshev functionals for different fractional integral operators. If we consider ρ = 1 in Theorem 2.1 and Theorem 2.2, then the obtained results will reduce to the said inequalities obtained by Dahmani et al. [9].