Some Inequalities Using Generalized Convex Functions in Quantum Analysis

Artion Kashuri,Miguel J Vivas-Cortez,Jorge E Hernández Hernández,Rozana Liko

doi:10.3390/sym11111402

Artion Kashuri, Miguel J Vivas-Cortez + Show 2 more

Open Access

https://doi.org/10.3390/sym11111402

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Abstract

In the present work, the Hermite–Hadamard inequality is established in the setting of quantum calculus for a generalized class of convex functions depending on three parameters: a number in ( 0 , 1 ] and two arbitrary real functions defined on [ 0 , 1 ] . From the proven results, various inequalities of the same type are deduced for other types of generalized convex functions and the methodology used reveals, in a sense, a symmetric mathematical phenomenon. In addition, the definition of dominated convex functions with respect to the generalized class of convex functions aforementioned is introduced, and some integral inequalities are established.

Highlights

The quantum calculus was initiated by Euler in the 18th century (1707–1783), it is known as calculus with no limits
The types of generalized convexity called (m, h1, h2 )−convexity [18] and convexity dominated by a (m, h1, h2 )−convex function g are of interest to this work
Following the steps of the excellent works presented by the aforementioned authors, the Hermite–Hadamard inequality for (m, h1, h2 )−convex functions is established, the concept of dominated convexity by a (m, h1, h2 )−convex function g is introduced, and some integral inequalities involving other types of generalized convexity are deduced

Summary

Introduction

The quantum calculus was initiated by Euler in the 18th century (1707–1783), it is known as calculus with no limits. The types of generalized convexity called (m, h1 , h2 )−convexity [18] and convexity dominated by a (m, h1 , h2 )−convex function g are of interest to this work. Hermite–Hadamard inequality and its variant forms are useful for quantum physics where lower and upper bounds of natural phenomena modelled and described by integrals are frequently required [19,20]. Following the steps of the excellent works presented by the aforementioned authors, the Hermite–Hadamard inequality for (m, h1 , h2 )−convex functions is established, the concept of dominated convexity by a (m, h1 , h2 )−convex function g is introduced, and some integral inequalities involving other types of generalized convexity are deduced. The methodology used reveals, in a sense, a symmetric mathematical phenomenon

Preliminaries

About Generalized Convexity

Main Results

Quantum Hermite–Hadamard Inequality for Generalized Convex Functions

Conclusions