Some new Hermite\u2013Hadamard type inequalities for s-convex functions and their applications

Serap Özcan,İmdat İşcan

doi:10.1186/s13660-019-2151-2

Abstract

In this paper, we establish some new integral inequalities of Hermite–Hadamard type for s-convex functions by using the Hölder–İşcan integral inequality. We also compare our new results with the known results and show that the results which we obtained are better than the known results. Finally, we give some applications to trapezoidal formula and to special means.

Highlights

The classical or the usual convexity is defined as follows:A function f : I ⊆ R → R is said to be convex on interval I if f tx + (1 – t)y ≤ tf (x) + (1 – t)f (y) for all x, y ∈ I and t ∈ [0, 1]
The following inequality is known in the literature as the Hermite–Hadamard inequality for convex functions: f a + b ≤ 1 b f (x) dx ≤ f (a) + f (b)
In [8], Hudzik and Maligranda considered the class of s-convex functions in second sense

Summary

Introduction

The classical or the usual convexity is defined as follows:A function f : I ⊆ R → R is said to be convex on interval I if f tx + (1 – t)y ≤ tf (x) + (1 – t)f (y) for all x, y ∈ I and t ∈ [0, 1]. The following inequality is known in the literature as the Hermite–Hadamard inequality for convex functions: f a + b ≤ 1 b f (x) dx ≤ f (a) + f (b) . This is defined as follows: A function f : [0, ∞) → R is said to be s-convex in second sense if the inequality f tx + (1 – t)y ≤ tsf (x) + (1 – t)sf (y) holds for all x, y ∈ [0, ∞), t ∈ [0, 1], and s ∈ A number of studies have shown that many of the results obtained about the theory of inequalities has a close relationship with the theory of convex functions.

Results

Conclusion