Some weighted integral inequalities for differentiable preinvex and prequasiinvex functions with applications

Muhammad Amer Latif,Sever Silvestru Dragomir

doi:10.1186/1029-242x-2013-575

Abstract

In this paper, we present weighted integral inequalities of Hermite-Hadamard type for differentiable preinvex and prequasiinvex functions. Our results, on the one hand, give a weighted generalization of recent results for preinvex functions and, on the other hand, extend several results connected with the Hermite-Hadamard type integral inequalities. Applications of the obtained results are provided as well. MSC: 26D15, 26D20, 26D07.

Highlights

Let f : I ⊆ R → R be a convex mapping and a, b ∈ I with a < b
Suppose that f : K → R is a differentiable mapping on K such that f ∈ L([a, a + η(b, a)])
Suppose that f : K → R is a differentiable mapping on K and w : [a, a +

Summary

Introduction

Let f : I ⊆ R → R be a convex mapping and a, b ∈ I with a < b. If the mapping |f | is a quasi-convex function on [a, b], the following inequality holds:. Is a quasi-convex function on [a, b], for p > , the following inequality holds: b–a a p (p + ) p p max f (a) p– , f a+b p–. Every quasi-convex function is prequasiinvex with respect to the map η(v, u) = v–u, but the converse does not hold; see, for example, [ ]. In the present paper we give new inequalities of Hermite-Hadamard for functions whose derivatives in absolute value are preinvex and prequasiinvex Our results extend those results presented in very recent results from [ , , , ] and [ ] and generalize those results from [ , ] and [ ].

Setting x

In what follows we use the notations