Opial integral inequalities for generalized fractional operators with nonsingular kernel

Abstract

We consider the well-known classes of functions $\mathcal{U}_{1}(\mathbf{v},\mathtt{k})$ U 1 ( v , k ) and $\mathcal{U}_{2}(\mathbf{v},\mathtt{k})$ U 2 ( v , k ) , and those of Opial inequalities defined on these classes. In view of these indices, we establish new aspects of the Opial integral inequality and related inequalities, in the context of fractional integrals and derivatives defined using nonsingular kernels, particularly the Caputo–Fabrizio (CF) and Atangana–Baleanu (AB) models of fractional calculus.

Fractional Hermite–Hadamard–Fejer Inequalities for a Convex Function with Respect to an Increasing Function Involving a Positive Weighted Symmetric Function

“…In our present investigation, we have established new fractional HHF integral inequalities involving the weighted fractional operators associated with positive symmetric functions. The HHF fractional integral inequality (7) has been applied to other class of convex functions, such as p-convex functions [39], generalized convex functions [40], (η 1 , η 2 )-convex functions [41] and many others that can be found in the literature. Thus, the results obtained here can be also be applied to the above class of convex functions.…”

“…Whichever one we study, we can apply it to the other one; see, e.g., [1]. There are plenty of well-known integral inequalities that have been established for the convex functions (1) in the literature; for example, Ostrowski type integral inequalities [2], Simpson type integral inequalities [3], Hardy type integral inequalities [4], Olsen type integral inequalities [5], Gagliardo-Nirenberg type integral inequalities [6], Opial type type integral inequalities [7,8] and Rozanova type integral inequalities [9]. However, the most common integral inequalities are the Hermite-Hadamard type integral inequalities: the classical and fractional Hermite-Hadamard type integral inequalities [10,11] are, respectively, given by:…”

Hermite–Jensen–Mercer type inequalities for conformable integrals and related results

“…Fractional calculus, one of the areas where inequality theory has benefited most in recent years, is an area that continues its development with a high acceleration by defining new fractional derivative and integral operators. Operators' applications in various fields, such as economics, applied mathematics, engineering, and mathematical biology, add strength to fractional analysis (see [19][20][21][22][23][24][25][26]). Now, we recall the definition of conformable integral of arbitrary order including the higher order case, on which our proven inequalities will be based.…”

Midpoint Inequalities in Fractional Calculus Defined Using Positive Weighted Symmetry Function Kernels

“…Most of these versions are described in the RL sense based on the corresponding fractional integral. Many integer-order integral inequalities such as Ostrowski [17], Simpson [18], Hardy [19], Olsen [20], Gagliardo-Nirenberg [21], Opial [22,23] and Rozanova [24] have been generalized and reformulated from the fractional point of view.…”