Fractional Integral Inequalities via Atangana-Baleanu Operators for Convex and Concave Functions

Abstract

Recently, many fractional integral operators were introduced by different mathematicians. One of these fractional operators, Atangana-Baleanu fractional integral operator, was defined by Atangana and Baleanu (Atangana and Baleanu, 2016). In this study, firstly, a new identity by using Atangana-Baleanu fractional integral operators is proved. Then, new fractional integral inequalities have been obtained for convex and concave functions with the help of this identity and some certain integral inequalities.

Ostrowski-type inequalities pertaining to Atangana–Baleanu fractional operators and applications containing special functions

“…They introduced some modified ABC fractional operators and also solved related differential equations. Many researchers [43][44][45][46][47] have studied the fractional integral, which Atangana and Baleanu [25] generalized. The corresponding derivative operator in the Caputo and Liouville-Reimann senses is Definition 1.6 ([25]) Let q > p, ς ∈ [0, 1] and Q ∈ H 1 (p, q).…”

On new general inequalities for s-convex functions and their applications

“…The convex functions theory plays an important role in all the fields of pure and applied mathematics. Some noteworthy inequalities have been acquired using the different types of convexity [7][8][9]. One of these types of convexity is s-convexity.…”

Analysis of Fractional-Order System of One-Dimensional Keller–Segel Equations: A Modified Analytical Method

“…In recent generations, fractional derivatives have been used to mathematically explain a variety of physical challenges, and these interpretations have shown good results when used to mimic real-world circumstances. Hadamard, Riemann-Liouville, Coimbra, Grunwald-Letnikov, Riesz, Weyl, Liouville Caputo, Atangana-Baleanu, Caputo-Fabrizio and some others, provided essential definitions of fractional operators [1][2][3][4][5]. To analyse the nonlinear FPDE solutions, several sophisticated approaches for discovering precise solutions have been devised, such as the Hermite colocation method [6], the optimal homotopy asymptotic technique [7], the Adomian decomposition method [8], the homotopy perturbation transform method [9], the Pade approximation and homotopy-Pade technique [10], the invariant subspace method [11], the q-homotopy analysis transform method [12], the homotopy analysis Sumudu transform method [13] and the Sumudu transform series expansion method [14].…”