Grüss type inequalities via generalized fractional operators

Abstract

One of the main motivation points in studies on inequalities is to obtain generalizations and to introduce new approaches. In this direction, the generalized fractional integral operators defined within the scope of fractional analysis are quite functional. In this paper, some new integral inequalities have been proved by using generalized fractional integral operators and some classical inequalities for integrable functions. In the proofs of the main findings, the definitions of the generalized fractional integral operator, certain classical relations, and some classical inequalities are used.

Certain New Chebyshev and Grüss-Type Inequalities for Unified Fractional Integral Operators via an Extended Generalized Mittag-Leffler Function

“…Let Ω(x) = 1, φ(x) = x λ/k and ξ(x) = x 1+β /(1 + β) for λ, k > 0, β = −1, then operators in (143) and (144) reduce to the generalized (k, β)-fractional integrals β F σ,k ρ,λ,a + ψ(x) and β F σ,k ρ,λ,b − ψ(x) defined by Tunc et al [63] and Butt et al [64]. (G6)…”

Some New Fractional Integral Inequalities Pertaining to Generalized Fractional Integral Operator

“…The extended generalized Mittag-Leffler function was used by Akdemir [28] to investigate several Grüss-type integral inequalities for fractional integral operators. Akdemir [29] also used the generalized fractional integral operator to analyze several Grüss-type inequalities. Using the generalized Katugampola fractional integral operator, Aljaaidi [30] investigated and proved various Grüss-type inequalities.…”

Pólya–Szegö Integral Inequalities Using the Caputo–Fabrizio Approach

“…In [29], the authors defined the weighted Caputo-Fabrizio fractional derivative and studied related linear and nonlinear fractional differential equations. In the literature, very little work has been reported on fractional integral inequalities using [33,34] proved some new integral inequalities by using generalized fractional integral operators and some classical inequalities for integrable functions and their applications to the Zipf-Mandelbrot law. Motivated by the above work, the main objective of this article is to establish some new results for the Pólya-Szegö inequality and some other inequalities using the Caputo-Fabrizio fractional integrals.…”