On a new class of convex functions and integral inequalities

Shanhe Wu,Muhammad Uzair Awan,Khalida Inayat Noor,Sabah Iftikhar,Muhammad Aslam Noor

doi:10.1186/s13660-019-2074-y

Abstract

The aim of this paper is to introduce a new extension of convexity called σ-convexity. We show that the class of σ-convex functions includes several other classes of convex functions. Some new integral inequalities of Hermite–Hadamard type are established to illustrate the applications of σ-convex functions.

Highlights

A set K ⊂ R is said to be convex if ∀x, y ∈ K, t ∈ [0, 1], we have (1 – t)x + ty ∈ K.A function f : K → R is said to be convex in the classical sense, if ∀x, y ∈ K, t ∈ [0, 1], we have f (1 – t)x + ty ≤ (1 – t)f (x) + tf (y).The theory of convexity plays a vital role in different fields of pure and applied sciences
The classical concepts of convex sets and convex functions have been generalized in different directions
Many famous inequalities can be obtained using the concept of convex functions

Summary

Introduction

A set K ⊂ R is said to be convex if ∀x, y ∈ K, t ∈ [0, 1], we have (1 – t)x + ty ∈ K.A function f : K → R is said to be convex in the classical sense, if ∀x, y ∈ K, t ∈ [0, 1], we have f (1 – t)x + ty ≤ (1 – t)f (x) + tf (y).The theory of convexity plays a vital role in different fields of pure and applied sciences. This result of Hermite and Hadamard reads as follows: Theorem 1.1 Let f : [a, b] ⊂ R → R be an integrable convex function.

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Conclusion