Abstract
In the article, we establish serval novel Hermite–Jensen–Mercer-type inequalities for convex functions in the framework of the k-fractional conformable integrals by use of our new approaches. Our obtained results are the generalizations, improvements, and extensions of some previously known results, and our ideas and methods may lead to a lot of follow-up research.
Highlights
Proof It follows from Lemma 2.5 and Jensen–Mercer inequality using the convexity of |τ | that
1 τ (x) q + 1 τ (y) q for all x, y ∈ [θ, θ]. Proof It follows from Lemma 2.5, Jensen–Mercer inequality, the convexity of |τ |q, and Hölder–İşcan integral inequality given in Theorem 1.4 of [70] that
3β, +1 αk holds for all x, y ∈ [θ, θ]. Proof It follows from Lemma 2.5, Jensen–Mercer inequality, the convexity of |τ |q, and the improved power-mean integral inequality given in Theorem 1.5 of [70] that
Summary
Convex function [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] is an important concept that has come to the fore among many other function classes with its many features and areas of use. The following Theorem 1.1 for convex functions can be found in [32]. Theorem 1.1 ([32]) Let τ be a convex function defined on [θ , θ].
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