Generalization of cyclic refinements of Jensen\u2019s inequality by Fink\u2019s identity

Nasir Mehmood,László Horváth,Josip Pečarić,Saad Ihsan Butt

doi:10.1186/s13660-018-1640-z

Abstract

We generalize cyclic refinements of Jensen’s inequality from a convex function to a higher-order convex function by means of Lagrange–Green’s function and Fink’s identity. We formulate the monotonicity of the linear functionals obtained from these identities utilizing the theory of inequalities for n-convex functions at a point. New Grüss- and Ostrowski-type bounds are found for identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity and mean value theorems.

Highlights

1 Introduction Arising from the circumstantial analysis of geometrical observations, the theory regarding convex functions serves to aid us in topics such as real analysis and economics
The theory of convex functions has progressed to quite a substantial extent
Several reasons may be attributed to this development: firstly, the application of convex functions is linked, directly or indirectly, to many fields of modern analysis; secondly, convex functions are deeply associated with the philosophy of inequalities and vice versa

Summary

Introduction

Arising from the circumstantial analysis of geometrical observations, the theory regarding convex functions serves to aid us in topics such as real analysis and economics. Theorem A (Classical Jensen’s inequality, see [8]) Let g be an integrable function on a probability space (X, A, μ) taking values in an interval I ⊂ R. We obtain generalizations of discrete and integral Jensen-type linear functionals with real weights. Pečarić et al in [12] study necessary and sufficient conditions on two linear functionals Ω : C([α1, d]) → R and Δ : C([d, α2]) → R so that the inequality Ω(φ) ≤ Δ(φ) holds for every function φ that is n-convex at point d. We are ready to state the following theorem for inequalities involving n-convex function at a point. Where Fαα (ξ , ·), GL(·, r) are the same as defined in (2), (3), respectively, and let Ωi(φ[α1,d]) and Δi(φ[d,α2]) be the linear functionals given by (23) and (24). If φ : [α1, α2] → R is (n + 1)-convex at point d, (36)

Fix i

By denoting