A new generalization of some quantum integral inequalities for quantum differentiable convex functions

Yi-Xia Li,Hüseyin Budak,Muhammad Aamir Ali,Yu-Ming Chu,Mujahid Abbas

doi:10.1186/s13662-021-03382-0

Abstract

In this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results.

Highlights

The study of q-calculus was initiated in the early 20th century after the work of Jackson (1910) who defined an integral later known as the q-Jackson integral
Applications of q-calculus can be found in various disciplines of mathematics and physics
The purpose of this paper is to prove several new quantum integral inequalities by applying the newly defined concept of a qb-integral

Summary

Introduction

The study of q-calculus was initiated in the early 20th century after the work of Jackson (1910) who defined an integral later known as the q-Jackson integral (see [16, 22, 23, 27, 28]). For more discussion on this subject, we refer to [8, 21]. Many well-known integral inequalities, such as Hölder inequality, Hermite–Hadamard inequalities and Ostrowski inequality, Cauchy–Bunyakovsky–Schwarz inequality, Grüss inequality, Grüss–Chebysev inequality, and other integral inequalities, have been studied in the setup of q-calculus using the concept of classical convexity. For more results in this direction, we refer to [1,2,3,4,5,6,7, 10, 11, 14, 18,19,20, 24, 29,30,31,32, 34, 35, 37, 39,40,41,42,43, 46, 47]

Objectives

Conclusion