Hermite-Hadamard and Schur-Type Inequalities for Strongly -Convex Fuzzy Interval Valued Functions

Abstract

The concept of fuzzy theory was developed in 1965 and becomes an acknowledged research subject in both pure and applied mathematics and statistics, showing how this theory is highly applicable and productive in many applications. In the present study, we introduced the definition of fuzzy interval valued strongly -convex function and investigated some of its properties. We established Hermite-Hadamard and Schur-type inequalities for the class of fuzzy interval valued strongly -convex function.

1. Introduction

The theory of interval analysis is one of the top areas of research nowadays because of its enormous application in various fields especially in automatic error analysis [1], computer graphics [2], and neural network output optimization [3]. In [4], Moore et al. give the first monograph on interval analysis, and since then, a huge amount of work have been done in interval calculus; for example, in [5], Chalco Cano et al. studied the interval-valued functions using generalized Hukuhara derivative and presented applications of interval valued calculus. An efficient method for solving fuzzy optimization problems using interval valued calculus is presented in [6]. Costa et al. [7] calculated the possible conformations arising from uncertainty in the molecular distance geometry problem using constraint interval analysis. Interval vector spaces and interval optimizations are given in [8], while optimality conditions for generalized differentiable interval valued functions are presented in [9]. Several classical inequalities for interval valued functions have been established in [10].

To address the modern problems, the convexity has been generalized in number of ways and some interesting generalizations are strongly -convex functions [11], strongly generalized convex functions [12], -convex function [13], and many more. The authors introduce -convex function in [14] as follows;

A function is known as -convex if for all and .

The authors introduce the strongly convex function in [15] as follows:

A function with modulus is defined as for all and which is called strongly convex.

The unification of above two definitions is defined in [16] as follows:

A function with modulus defined as follows: for all and which is known as strongly -convex.

In recent years, mathematical inequalities for interval valued convex and nonconvex function got attention of many mathematician [17]. Bai et al. [10] established Hermite-Hadamard and Jensen-type inequalities for the class of interval valued nonconvex functions. In 2017, Costa [18] presented Jensen-type inequality for the class of fuzzy interval valued functions, and in the same year, Costa and Román–Flores [19] presented some integral inequalities for the same class of functions. Some Opial–type inequalities were studied in [20]. For other remarkable results, we refer [21] and references therein.

In this report, we proposed the definition of fuzzy interval valued strongly -convex function. We investigated some of its properties and established Hermite-Hadamard and Schur-type inequalities for the proposed definition.

The paper is organized as follows: In Section 2, we will give some preliminaries and basic definitions, and we will established some basic properties. However, Section 3 is devoted for the establishment of main results like Hermite Hadamard and Schur-type inequalities.

2. Some Basic Properties of Interval Calculus

Let and be all positive intervals and all intervals. The algebraic operations “+” and “.” are defined as [22] and

A function with , where are real functions and known as interval valued functions if , for all .

For intervals and , the distance is the Hausdorff distance. Then is complete.

The tagged partition of interval is the set of numbers if and if , for all .

Moreover, letting and for every , the partition is said to be -fine. The family of the all such -fine partitions of is represented as [17].

Let be the -fine partitions of ; then the function is said to be integral sum if

Proposition 1 (see [17]). The four arithmetic operators for and are defined as where .

Definition 2. A function is called convex function if for all and .

Definition 3 (see [23]). Let a nonnegative function , and . We say that a function is interval -convex, if for all and , we have If the set inclusion (9) is reversed, then is said to be interval -concave, i.e., .
We end this section of preliminaries by introducing the new concept of interval valued strongly -convex function. Note that for interval and , the inclusion is defined by

Definition 4. Let be a normed space and is convex. is a function. We call a function with modulus is interval valued strongly -convex function if for all and .

The notion is reserved for the class of interval valued strongly -convex functions.

Remark 5. (1)By taking , then the set inclusion (11) reduces to interval valued -convex function [23](2)By taking , then the set inclusion (11) reduces to strongly -convex function [16](3)By taking and , then the set inclusion (11) reduces to -convex function [14](4)By taking and , then the set inclusion (11) reduces to strongly-convex function [15](5)By taking , and , then the set inclusion (11) reduces to classical convexity(6)By taking , and , then the set inclusion (11) reduces to Godunova- Levin function

3. Main Results

In this section, we establish the Hermite-Hadamard- and Schur-type inequalities for the proposed definition.

Theorem 6. Let be a normed space and is convex. is a function with holding condition for each if belongs to ; then is defined as which also belongs to .

Proof. Consider ; then which gives the desired result.

Theorem 7. Let be a normed space and is convex. is a function with holding condition for each if belongs to ; then is defined as which also belongs to .

Proof. The proof is similar to that of Theorem 6.

Example 8. Let , . Then defined by , is interval valued strongly -convex with modulus Because for every and , then from (11), we have Hence, the inequalities (15) and (16) hold for , , and . So the set inclusion (11) holds, and we say that is interval valued strongly -convex function.
But it is not interval valued -convex function. Because by taking , , , , and , the inequality (16) is not satisfied. So the set inclusion (9) is holds, and we say that it is not interval valued -convex function.

3.1. Interval Hermite-Hadamard-Type Inequality

Theorem 9. Let a nonnegative function . A function defined over where is interval valued strongly -convex function with modulus ; then

Proof. Take , , and and applying the definition 4, we obtained which implies that Substituting the values of and in (21), we get integrating (23) with respect toover; we obtain Similarly substituting the values of and in (22), we get Combining (24) and (25), we obtain which gives the proof of left hand side.
For the proof of right hand side, take Using the definition of interval valued strongly -convex function, we obtain which gives right hand side.

Combining (26) and (28), we obtain desire result.

Remark 10. (1)By taking , then (17) becomes Hermite Hadamard-type inequality for interval valued h-convex function [23](2)By substituting , then (17) becomes Hermite Hadamard-type inequality for strongly h-convex function [16](3)By substituting and , then (17) becomes Hermite Hadamard-type inequality for strongly convex function [15](4)For , and , then (17) becomes classical Hermite Hadamard-type inequality

Example 11. Consider for , , and be defined by and ; then, we have Also, Returning to (29), (30), and (31), we deduce

Consequently, Theorem 9 is verified.

3.2. Interval Schur-Type Inequality

Theorem 12. Consider be the interval valued strongly -convex function and a nonnegative super multiplicative function for all and such that ; then the following inequality holds: where iff is interval valued strongly h-convex function.

Proof. Let be numbers which satisfy assumptions of the proposition. Then , , and Also, Assume that and is interval valued strongly convex function with modulus . Let ; then substituting , , and in (11) yields that implying Now, we write the above set inclusion as Using the condition (35), we obtain Now, we write the above inequalities in set inclusion form as required. Conversely suppose that (33) holds, and we insert , , and in (33); we get Use condition (35), and simplify the above inclusion set yields that is interval valued strongly -convex, which completes the proof.

Remark 13. If we take and in Theorem 12, then, we obtain Schur-type inequality for -convex function [24].

4. Concluding Remarks

In this report, we introduced the fuzzy interval valued strongly -convex function. We also investigated some of its properties. Moreover, we established Hermite-Hadamard and Schur-type inequalities for the proposed definition. Our results are more general than many existing results. It is interesting for the researcher to find other types of inequalities for the proposed class of functions in the setting of various fractional integrals.

Data Availability

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

Conflicts of Interest

The authors do not have any competing interests.

Authors’ Contributions

Putian Yang proved the results and arranged the funding for this paper, and Shiqing Zhang wrote the paper and analyzed the results.

Acknowledgments

This research is funded by the Department of Mathematics, Sichuan University, Chengdu 610064, China.