Some Inequalities Related to Interval-Valued -Convex Functions

Abstract

Convexity plays an important role in many areas of mathematics, especially in the study of optimization problems where they are distinguished by a number of convenient properties. Our aim is to introduce a more extended version of convexity. In this paper, we introduced interval-valued generalized convex function and proved Hermite–Hadamard-, Jensen-, and Ostrowski-type inequalities in this generalization. The presented results are generalizations of many existing results of literature.

1. Introduction

In nonlinear programming and optimization theory, convexity plays an important role. There were three important areas of nonlinear analysis: monotone operator theory, convex analysis, and theory of nonexpensive mapping, but in early 1960, these theories get emerged. These areas got the attention of many researchers, and many connections have been identified between them over the past few years.

It is always interesting and appreciable to generalize the definition of convexity from different aspects because it helps to tackle modern world problem. The famous generalizations of convexity are E-convex functions [1], convex functions [2], -convex function [3], and convex vector [4].

Consider a convex function , then the following inequality holds:where and .

In literature, we can find many generalization of inequality (1) for different classes of convex functions, for example, Delavar and Dragomir [5] generalized (1) for -convex functions and Varosanec in [6] generalized (1) for convexity. For other generalizations, we refer the readers to [7–9].

In 1996, Moore in [10] introduced the interval analysis. Later on, Costa et al. [11] introduced generalized interval vector spaces and interval optimization. The convex functions in interval-valued calculus got attention of researchers due to its interesting geometric features and infimum properties [12–15]. This theory is also appealing for the applied engineers and programmes due to its applications in convex optimization [16–18]. To increase the reader’s interest in convex analysis and interval analysis, we recommend [6, 7, 19–25].

In this paper, we introduced generalized convex function over interval-valued setting and proved Hermite–Hadamard-, Jensen-, and Ostrowski-type inequalities. The paper is organized as follows: in Section 2, we present some basic notions and definitions. Section 3 contains some basic properties interval-valued generalized convex functions, and Section 4 contains Hermite–Hadamard-, Jensen-, and Ostrowski-type inequalities for interval-valued generalized convex functions. The paper is concluded in Section 5.

2. Preliminaries

Throughout this paper, and denote the family of all intervals and positive intervals equipped with the algebraic operations “+” and “.,” respectively. The algebraic operations “+” and “.” are defined by

Let are real functions such that for all , then the function defined byis said to be an interval-valued function.

For any two intervals and , the Hausdorff distance is defined byand is called Hausdorff space.

A set of numbers is tagged partition of ifand if for all . Moreover, if we let and for each , then the partition is -fine. The family of all -fine partitions of is represented by .

Given , we define an integral sum of as follows:

The four arithmetic operators (+, −, ., /) for and are defined as follows:(1).(2).(3).(4).

Here,

Now, we present some definitions and introduce interval-valued generalized convex functions. From now to onward in this section, we take .

Definition 1 (convex function [26]). A function is said to be convex, iffor all and .

Definition 2 (-convex function [5]). A function is said to be convex with respect to for appropriate iffor all and .

Example 1. Consider a function defined byDefinefor all , then is an -convex function but not convex.

Definition 3 (-convex function [6]). Let is a nonnegative function, where and . A function is said to be -convex if is nonnegative andfor all and .
The class of -convex functions is denoted by .

Definition 4 (modified -convex function [27]). Let be a nonnegative function, where and . A function is said to be modified -convex iffor all and .

Definition 5 (supermultiplicative function [28]). A function is said to be supermultiplicative iffor all .

Definition 6 (-convex function [28]). Let be a nonnegative function, where and . A function is said to be -convex iffor all and .

Example 2. Consider a function defined byDefine a bifunction as , for all and , where . Then, is an -convex function but not convex.

Definition 7 (interval-valued -convex function [29]). Let be a nonnegative function, where and . A function is said to be interval-valued -convex, if is nonnegative andfor all and .
The class of interval-valued -convex functions is denoted by .

Definition 8 (P-function [30]). A function is said to be -function if is nonnegative andfor all and .
Now, the definition of interval valued -convex function is as follows.

Definition 9. (interval-valued generalized -convex function). Let be a nonnegative function, where and . Also, let be a bifunction for appropriate . A function is said to be interval-valued generalized convex iffor all and .

Remark 1. Of course, (18) is -convexity for , modified -convexity for , and classical convex function for and simultaneously.

Remark 2. If we take in (18), we getfor any and .

Remark 3. If we take and in (18), we getfor any . The second condition obviously implies the first.

3. Basic Results

In this section, we proved some basic results.

Remark 4. (operation which preserves generalized convexity). Let are two generalized convex functions. Then, the following statements hold:(1)If is additive, then is generalized convex.(2)If is nonnegatively homogeneous, then for any , is generalized convex.

Theorem 1. Let and are two nonnegative functions defined on with the property . If is interval valued -convex function, then is interval valued -convex function.

Proof. For any and , we havewhich implies that is interval-valued -convex function.

Theorem 2. Let be an interval-valued function such that . Then, , iff and .

Proof. Suppose that , for any and , then we havethat is,It follows thatThis shows that and.
Conversely, suppose that and , then from Definition 8 and inclusion (23), it follows that .

Theorem 3. Let be an interval valued function such that , then iff and .

Proof. The proof is similar to that of Theorem 2, so we omit it.

4. Main Results

Theorem 4. (Hermite–Hadamard-type inequality). Let be an interval valued generalized convex function such that is bounded above on , i.e., . Then, we have

Proof. Take and , then we haveSo,By definition of interval-valued generalized convex functions, we havewhich follows thatIntegrating (29) with respect to on , we getNow,Similarly,Integrating (30) with respect to on , we haveNow,impliesSimilarly,Adding (32) and (33), we getAdding (36) and (37), we getCombining (31) and (38), we getCombining (34) and (39), we getCombining (40) and (41), we getwhich leads us to our desired result (25).

Remark 5. If we choose , , and in (25), we get the classical Hermite–Hadamard-type inequality for convex function.

Example 3. Suppose and for all . Furthermore, suppose be defined byfor all . Then, we havewhich impliesFurthermore,Combining (45), (46), and (47), we haveConsequently, we get Theorem 4.

Example 4. Suppose for . Also, suppose be defined by and be an odd number, then we havewhich impliesCombining (49), (50), and (51), we haveConsequently, we get Theorem 4.

Theorem 5. (Jensen-type inequality). Let be an -convex function. Then, for and , we haveAlso, for , and , when , we have

Theorem 6. Let be an -convex and be nondecreasing, nonnegatively, and sublinear in first variable. If for such that , thenwhere and for all .

Proof. Since is nondecreasing, nonnegatively, and sublinear in the first variable, from (55), it follows thatNow,From (56) and (57), we have (55).

Remark 6. If we choose , , and in (55), then we have the Jensen inequality derived in [28].

Lemma 1. Let be a differentiable mapping on , where and . If , then the following equality holds:for each .

Theorem 7. (Ostrowski-type inequality). Let be a nonnegative function. Also, let be a differentiable mapping on such that with and with . If is an -interval-valued convex function on and for all , then we havewhereand .

Proof. Since is interval-valued convex function, so by using Lemma 1, we can writeAlso,Moreover, we haveFurthermore, we haveBy combining (64) and (66), we getThis gives our desired result (59).

Remark 7. If we choose , and in (67), then we get the classical Ostrowski inequality for convex functions [31].

5. Conclusion

Convexity and interval analysis are mathematical tools that help to solve real-world problems more accurately; that is why, several generalizations of interval valued convex functions are the main objective of mathematicians. In this paper, we introduced generalized interval-valued convex functions and presented different types of well-known inequalities for this proposed class of convex functions. It is worthy to mention here that our results are the extensions of many existing results (for example, [28, 31, 32]).

Data Availability

All data required for this research are included within this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally in this paper.

Acknowledgments

The work was partially funded by the University of Okara, Pakistan.