Some New Estimates on Coordinates of Left and Right Convex Interval-Valued Functions Based on Pseudo Order Relation

Muhammad Bilal Khan,Pshtiwan Othman Mohammed,Yasser S Hamed,Kamsing Nonlaopon,Hari Mohan Srivastava

doi:10.3390/sym14030473

Muhammad Bilal Khan, Pshtiwan Othman Mohammed + Show 3 more

Open Access

https://doi.org/10.3390/sym14030473

Copy DOI

Abstract

The relevance of convex and non-convex functions in optimization research is well known. Due to the behavior of its definition, the idea of convexity also plays a major role in the subject of inequalities. The main concern of this paper is to establish new integral inequalities for newly defined left and right convex interval-valued function on coordinates through pseudo order relation and double integral. Some of the Hermite–Hadamard type inequalities for the product of two left and right convex interval-valued functions on coordinates are also obtained. Moreover, Hermite–Hadamard–Fejér type inequalities are also derived for left and right convex interval-valued functions on coordinates. Some useful examples are also presented to prove the validity of this study. The proved results of this paper are generalizations of many known results, which are proved by Dragomir, Latif et al. and Zhao, and can be vied as applications of this study.

Highlights

Because the functions discovered in a large number of theoretical and practical economics problems are not classical convex functions, many scholars have been interested in the sweeping generalisation of function convexity in past few decades, such as h-convex functions [1–5], log-convex functions [6–9], log-h-convex functions [10], and especially coordinated convex functions [11]
Moore’s interval analysis theory, which he proposed in a numerical analysis in [18], has advanced rapidly in recent decades
Interval analysis is commonly used in chemical and structured engineering, economics, control circuitry design, robotics, beam physics, behavioural ecology, constraint satisfaction, computer graphics, signal processing, asteroid orbits and global optimization and neural network output optimization [19], and many other fields

Summary

Introduction

Convex analysis has made major contributions to the improvement of various fields of applied and pure study. Many authors have proposed different expansions and generalizations of integral inequalities for coordinated convex functions since 2001 (see [12–17] and the references therein). We noted that most of authors used inclusion relation to obtain different types of inequalities for interval-valued functions, such as Zhao et al [24] who, in 2008, developed h-convex I-V-Fs (h-convex I-V-Fs) and demonstrated the following. I-V-Fs and establishing interval-valued Hermite–Hadamard type inequality for (h1 , h2 )convex I-V-Fs. We suggest that readers consult [26–28] and the references therein for more examination of the literature on the applications and properties of generalized convex functions and HH type integral inequalities. Zhang et al [29] introduced pseudo order relation on the space of interval and proposed the new class of convex functions in interval-valued settings by using pseudo order relation, which is known as left and right convex I-V-Fs

Preliminaries

Hermite–Hadamard Inequalities on Coordinates

Conclusions