Abstract
There have been many different definitions of fractional calculus presented in the literature, especially in recent years. These definitions can be classified into groups with similar properties. An important direction of research has involved proving inequalities for fractional integrals of particular types of functions, such as Hermite–Hadamard–Fejer (HHF) inequalities and related results. Here we consider some HHF fractional integral inequalities and related results for a class of fractional operators (namely, the weighted fractional operators), which apply to function of convex type with respect to an increasing function involving a positive weighted symmetric function. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities.
Highlights
First of all, we recall the basic notation in convex analysis
There are plenty of well-known integral inequalities that have been established for the convex functions (1) in the literature; for example, Ostrowski type
We have established new fractional HHF integral inequalities involving the weighted fractional operators associated with positive symmetric functions
Summary
A set V ⊂ R is said to be convex if ε θ1 + (1 − ε)θ2 ∈ V for each θ1 , θ2 ∈ V and ε ∈ [0, 1]. Based on a convex set V , we say that a function } : V → R is convex, if the inequality. We say that } is concave if −} is convex. Theory and application of convexity play an important role in the field of fractional integral inequalities due to the behavior of its properties and definition, especially in the past few years. There is a strong relationship between theories of convexity and symmetry. There are plenty of well-known integral inequalities that have been established for the convex functions (1) in the literature; for example, Ostrowski type
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