Abstract
The goal of this present paper is to study some new inequalities for a class of differentiable functions connected with Chebyshev’s functionals by utilizing a fractional generalized weighted fractional integral involving another function mathcal{G} in the kernel. Also, we present weighted fractional integral inequalities for the weighted and extended Chebyshev’s functionals. One can easily investigate some new inequalities involving all other type weighted fractional integrals associated with Chebyshev’s functionals with certain choices of omega (theta ) and mathcal{G}(theta ) as discussed in the literature. Furthermore, the obtained weighted fractional integral inequalities will cover the inequalities for all other type fractional integrals such as Katugampola fractional integrals, generalized Riemann–Liouville fractional integrals, conformable fractional integrals and Hadamard fractional integrals associated with Chebyshev’s functionals with certain choices of omega (theta ) and mathcal{G}(theta ).
Highlights
The Chebyshev functional [1] for two integrable functions U and V on [v1, v2] is defined by v2 T(U, V) =U (θ)V(θ) dθ v1 – v2 v1 v1 – v2 U (θ) dθ v1 V(θ) dθ . (1.1)The weighted Chebyshev functional (WCF in short) [1] for two integrable functions U and V on [v1, v2] is defined by T(U, V, ) =(θ) dθ (θ)(θ)(θ) dθ (θ)(θ) dθ (θ)V(θ) dθ, (1.2)
5 Concluding remarks We presented some new weighted fractional integral inequalities for a class of differentiable functions connected with Chebyshev’s, weighted Chebshev’s and extended Chebyshev’s functionals by utilizing weighted fractional integral operator recently introduced by Jarad et al [45]
These inequalities are more general than the existing classical inequalities given in the literature
Summary
The Chebyshev functional [1] for two integrable functions U and V on [v1, v2] is defined by v2. Definition 2.6 ([45]) Let the function U be an integrable in Xp (0, ∞) and suppose the function G is positive, increasing and monotone on [0, ∞) and having continuous derivative on [0, ∞) such that G(0) = 0. Setting G(θ ) = ln θ in Definition 2.6, we get the following weighted Hadamard fractional integral operator: x1 Iωκ U (ln θ ln θ)κ–1ω(θ)U (θ) dθ v1 θ iii. Definition 2.7 Let the function U be an integrable in the space XUp (0, ∞) and suppose the function G is positive, increasing and monotone on [0, ∞) and having continuous derivative on [0, ∞) such that G(0) = 0. For all θ > 0, κ > 0, the following fractional integral inequality holds:
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