Certain Chebyshev-Type Inequalities Involving Fractional Conformable Integral Operators

Abstract

Since an interesting functional by P.L. Chebyshev was presented in the year 1882, many results, which are called Chebyshev-type inequalities, have been established. Some of these inequalities were obtained by using fractional integral operators. Very recently, a new variant of the fractional conformable integral operator was introduced by Jarad et al. Motivated by this operator, we aim at establishing novel inequalities for a class of differentiable functions, which are associated with Chebyshev’s functional, by employing a fractional conformable integral operator. We also aim at showing important connections of the results here with those including Riemann–Liouville fractional and classical integrals.

Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications

“…In [35], Aldhaifallah et al introduced some integral inequalities for a certain family of n(n ∈ N) positive continuous and decreasing functions on some intervals employing what is called generalized (k, s)-fractional integral operators. Recently, some researchers introduced a verity of certain interesting inequalities, applications, and properties for the conformable integrals [36][37][38][39][40].…”

Tempered Fractional Integral Inequalities for Convex Functions

“…Dahmani [47] presented some classes of fractional integral inequalities by considering a family of n positive functions. Certainly, remarkable inequalities such as Hermite-Hadamard type [48], Chebyshev type [49][50][51], inequalities via generalized conformable integrals [52], Grüss type [53,54], fractional proportional inequalities and inequalities for convex functions [55], Hadamard proportional fractional integrals [56], bounds of proportional integrals with applications [57], inequalities for the weighted and the extended Chebyshev functionals [58], certain new inequalities for a class of n(n ∈ N) positive continuous and decreasing functions [59] and certain generalized fractional inequalities [60] are recently presented by utilizing several different kinds of fractional calculus approaches.…”

The existence of solutions for nonlinear fractional boundary value problem and its Lyapunov-type inequality involving conformable variable-order derivative

“…The classical inequalities and their applications play an essential role in the theory of differential equations and applied mathematics. A large number of generalizations of classical inequalities by means of fractional operators are established in [9,10,[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] and the references therein. The Lyapunov inequality was presented and proved by a Russian mathematician A.M. Lyapunov in [30], where he stated the fact that if the boundary value problem (BVP)…”