Integral Inequalities of Hermite–Hadamard Type for Extended s-Convex Functions and Applications

Abstract

In the paper, the authors set up an identity for a function whose third derivative is integrable, establish by the Hölder inequality some new integral inequalities of the Hermite–Hadamard type for extended s-convex functions in the second sense, and apply these integral inequalities to construct inequalities for several special means.

Minkowski’s inequality for the AB-fractional integral operator

“…Jarad et al [16] presented a Gronwall-type inequality for the analysis of the fractional-order Atangana-Baleanu differential equation and in [17] for generalized fractional derivatives. [18] proved some Hermite-Hadamard-type inequalities for a class of s-convex functions and studied special means. Mehrez and Agarwal [19] produced new integral inequalities by means of classical Hermite-Hadamard inequalities and obtained particular cases of their results with applications to special means.…”

Refinements of Young's integral inequality via fundamental inequalities and mean value theorems for derivatives

“…When h (x) is strictly increasing, the double inequality (1.12) can be equivalently written as where J ⊆ R is a nonempty interval and τ, µ ∈ J with τ < µ. The double inequality (1.13) is called Hermite-Hadamard's integral inequality for convex functions [7,37,44]. When a > h −1 (b), as showed in Figures 3 and 5, and h (x) is strictly increasing, that is, the function h(x) is convex, as showed in Figures 3 Figure 6.…”

Notes on judgment criteria of convex functions of several variables

“…Due to the HermiteHadamard integral inequality, see the paper [9], the rst term in ( 4) is negative. On the other hand, by Taylor's formula, the second term in ( 4) is equal to…”