On Some Integral Inequalities Related to Hermite-Hadamard-Fejér Inequalities for Coordinated Convex Functions

Ruiyin Xiang,Feixiang Chen

doi:10.1155/2014/796132

Ruiyin Xiang, Feixiang Chen

Open Access

https://doi.org/10.1155/2014/796132

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Abstract

Several new mappings associated with coordinated convexity are proposed, by which we obtain some new Hermite-Hadamard-Fejér type inequalities for coordinated convex functions. We conclude that the results obtained in this work are the generalizations of the earlier results.

Highlights

Let f : I ⊆ R → R be a convex function and a, b ∈ I with a < b; f(a + 2 b ) ≤ 1 b−a b∫ f (t) dt a f (a) + f (b) 2 (1)is known as the Hermite-Hadamard inequality
Let f : [a, b] → R be a convex function, 0 < α < 1, 0 < β < 1, A = αa + (1 − α)b, u0 = (b − a) min{α/(1 − β), (1 − α)/β}, and let h be defined by h(t) = (1 − β)f(A − βt) + βf(A + (1 − β)t), t ∈ [0, u0]
Let f : [a, b]×[c, d] → R be a coordinated convex function, 0 < α < 1, 0 < β < 1, 0 < λ < 1, 0 < μ < 1, A = αa+ (1−α)b, B = λc+(1−λ)d, u0 = (b−a) min{α/(1−β), (1−α)/β}, V0 = (d − c) min{λ/(1 − μ), (1 − λ)/μ} and let h be defined by h (t, s) = (1 − β) (1 − μ) f (A − βt, B − μs)

Summary

Introduction

A function f : Δ → R is said to be convex on coordinates on Δ if the inequality f (λx + (1 − λ) z, ty + (1 − t) w) Dragomir in [5] established the following HermiteHadamard type inequalities for coordinated convex functions in a rectangle from the plane R2.

Results

Conclusion