Hermite-Hadamard- and Jensen-Type Inequalities for Interval Nonconvex Function

Abstract

In the present study, we will introduce the definition of interval nonconvex function. We will investigate some properties of interval nonconvex function. Moreover, we will develop Hermite-Hadamard- and Jensen-type inequalities for interval nonconvex function.

1. Introduction

Since its inception five decades ago, the theory of fuzzy sets has advanced in a variety of ways. Application of the theory of fuzzy sets covered many areas like artificial intelligence, decision theory, computer science, logic operational research, and robotics [1–3]. Initially good books like probability theory by Dubois and Prade in 1988, Behavioral and social science by Smithsons in 1987, and Fuzzy Control by sugeno 1985 and pedrycz 1989 and others have been published. We refer [4, 5] recent developments in this field.

For some other results and application of interval analysis theory, we refer the readers [4, 6–13]. Due to vast application of fuzzy sets, many integral inequalities have been derived by different authors [1–3, 14–17]

Costa [18] there is a new fuzzy version of Jensen-type integral inequality for fuzzy interval valued function. Also in [19], Zhao et al. develop new Harmite-Hadamard-type inequality for h-convex interval valued function.

For more about Hermite-Hadamard inequalities, refer [20–24]. We will introduce the interval non-convex function. The second objective of this article is to develop Hermite-Hadamard- and Jensen-type inequality for above said generalization.

2. Preliminaries

In this section, we define some basic definitions, properties, results, and notations on interval analysis, which are used throughout the paper [17, 25]. Here, and denote the family of all intervals and positive interval and it is equipped with the algebraic operations “+” and “.” given, respectively, by and

A function with , where are real functions with for all , it is called an interval-valued function.

For intervals and , the Hausdorff distance is defined by .

Then is complete.

A set of numbers is said to be a tagged partition of ifand if for all . Moreover, if we let and if for each , then we say that the partition is -fine. The family of all -fine partitions of is denoted by .

Given , we define a integral sum of as follows:

Throughout the paper, -integrable means interval Riemann integrable. The concept of -integrable is given in [19], Definition 2.2, is equivalent to integral given in [10], Definition 9.1.

Definition 1. Let . is called -integrable on with -integral , if there exists an such that for any there exists a such thatfor each . Let denote the set of all -integrable functions on .

Definition 2. (See -convex set, [26]).
The real interval is known as p-convex set if for all and , implies thatwhere or , , , and .

Definition 3. (See -convex function, [26]). For a -convex set , the mapping :is called -convex function, for all , and .

Definition 4. (See h-convex function, [27]). The nonnegative function is -convex if  and , is nonnegative real-valued function or belongs to the class .
If the inequality (7) is reversed, then is said to be -concave, i.e., .

Definition 5. -convex function [28]. Let be a nonnegative function, . The nonnegative function is an -convex iffor all and or that belongs to the class . If the inequality (8) is reversed, then is said to be -concave, i.e., .

Definition 6. Interval -convex function [29]. Let be a nonnegative function, . We say that is an interval -convex function or that belongs to the class , if is nonnegative and for all and , we haveIf the inequality (9) is reversed, then is said to be interval -concave, i.e., .

Remark 1. (1)If , then Definition 6 becomes interval h-convex in [19].(2)If , then Definition 6 becomes P-function in [30].(3)If , , then Definition 6 becomes to s-convex fuzzy process in [31].We wind up the current section by introducing the new concept of interval nonconvexity. This idea is inspired by An et al. [29]. Throughout the paper, for interval and ,

Definition 7. Interval nonconvex function. Let be a nonnegative function, . For a nonconvex function, is an interval nonconvex function if, and
For convenience, in this paper, the class of interval non-convex function is denoted by . If the inequality (11) is reversed, then is said to be interval nonconcave, i.e., .
Throughout the paper for. .

Theorem 1. Let be an interval-valued function such that . Then if and only if and .

Proof. Let be a interval non-convex function and suppose that , , thenthat is,It follows thatThis shows that and. .
Conversely suppose that
and then from above definition and inclusion set (13), it follows that .
This completes the proof.

Theorem 2. Let be an interval-valued function such that . Then if and only if and .

Proof. The proof is similar to that of Theorem 1.

3. Main Result

Theorem 3. Let , and if and then

Proof. By assumption, we haveIntegrating above w.r.t. “” on [0, 1], we getit follows thatThis implies thatNow by def. of interval nonconvex function, we haveintegrated with respect to “x” on [0, 1], we getit follows thatCombining (19) and (22) we get (27).

Remark 2. (1)If and then Theorem 3 becomes ([32], Theorem 5).(2)If and then Theorem 3 becomes ([19], Theorem 4.1).(3)If , and then Theorem 3 becomes ([31], Theorem 4).(4)If then Theorem 3 becomes Hermite-Hadamard inequality for -convex function.

Example 1. Consider for , and be defind by and be an odd number, then we haveandPutting in (24) and simplifying, we getCombining (23)–(25), we getConsequently, verify the Theorem 3.

Theorem 4. Let , , and if and thenwhere

Proof. For , one hasConsequently, we getSimilarly for Adding (30) and (31), we getand from Theorem 3, we haveand the result follows.

Theorem 5 (Jensen-type inequality). Let with . If is nonnegative super multiplicative function and if and . Then, we have following inequality:where

Proof. When n = 2 inequality (33) is true. Consider the inequality (33) is true for , thenThus, by mathematical induction the result is proved.

4. Conclusions

Hermite-Hadamard- and Jensen-type inequalities are true for this new concept of interval value convexity. Moreover by suitable substitution, the obtained results reduce to the results of [19, 31, 32].

Data Availability

All data required for this paper is included within this paper.

Conflicts of Interest

The authors do not have any conflicts of interest.

Authors’ Contributions

All authors contributed equally in this paper.

Acknowledgments

This research was supported by Higher Education Commission, Pakistan.