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Integral inequalities play critical roles in measure theory and probability theory. Given recent profound discoveries in the field of fuzzy set theory, fuzzy inequality has become a hot research topic in recent years. For classical Sandor type inequality, this paper intends to extend the Sugeno integral. Based on the (s,m)-convex function in the second sense, a new Sandor type inequality is proposed for the Sugeno integral. Examples are given to verify the conclusion of this paper.
Fuzzy sets are an important measurement tool for processing uncertain information. Fuzzy measurements and fuzzy integrals, which were originally introduced by Sugeno in 1974 , are important analytical methods of measuring uncertain information [2,3,4]. The Sugeno integral has been applied to many fields, such as management decision-making and control engineering [5,6,7,8,9]. Because of the special integral operator of the Sugeno integral, it is limited in many practical problems. To overcome this shortcoming, many scholars have replaced the Sugeno integral in large or small operations with new operators, and they have proposed various types of fuzzy integrals, such as the Shilkret integral , the bipolar level Choquet integral , the level-dependent Sugeno integral , etc. In recent years, some famous integral inequalities have been generalized to fuzzy integrals (cf. [13,14,15,16]). The study of inequalities for the Sugeno integral, which was initiated by Roman-Flores et al. , is the most popular.
The convexity method, i.e., establishing inequalities for convex functions, is one of the most powerful tools in establishing analytic inequalities. Particularly, there are many important applications in the study of higher transcendental functions . Convex functions have become the theoretical basis and a powerful tool in game theory, mathematical programming theory, economics, etc. Some famous convex functions have been put forward, and in recent years they have been used to establish integral inequalities. For example, Gill et al.  extended Hadamard’s Inequality on the basis of r-convex functions. Sarikaya and Kiris  established interesting and important inequalities based on Hermite-Hadamard type inequality for the s-convex in the second sense. Muddassar and Bhatti  established generalizations of Hadamard type inequalities through differentiability for the s-convex function. Micherda and Rajba  introduced a new class of (k,h)-convex functions defined on k-convex domains, and they proved new inequalities of Hermite-Hadamard and Fejér types for such mappings.
Sandor type inequality is an important integral inequality for convex functions. Most extended Sensor type inequalities are established on the basis of definite integrals, but the research based on fuzzy integral is still scarce [23,24,25,26,27]. Sándor  first introduced Sandor type inequality for definite integrals with respect to convex functions. Caballero and Sadarangani  extended Sandor type inequality for the Sugeno integral with convex functions. Based on other kinds of convex functions, for the Sugeno integral, other Sandor type inequalities have also been established. For example, Li et al.  derived Sandor type inequalities for the Sugeno integral with respect to general (α,m,r)-convex functions, Yang et al.  derived Sandor type inequality for the Sugeno integral with respect to (α,m) convex functions, and Lu et al.  obtained Sandor type inequality for the Sugeno integral with respect to r-convex functions. We find that various kind convex functions have different Sensor type inequalities. Studies of Sendor type inequalities can give good estimates of the Sugeno integral under various convex functions. (s,m)-Convex functions in the second sense are important convex functions, and an ordinary convex function is a special case . Many authors have been interested in achieving inequality for this function under definite integrals (see [29,30,31,32]). In this paper, we will extend the Sandor type inequality for (s,m)-convex functions in the second sense for the Sugeno integral. Some examples are given to illustrate the results. The organization of this article is as follows: Section 2 will recall the definitions and properties of the Sugeno integral and of the convex function. Section 3 will establish some new Sandor type inequalities for (s,m)-convex functions in the second sense based on the Sugeno integral. Finally, conclusions are drawn in Section 4.
In this section, some definitions and properties of the Sugeno integral and (s,m)-convex function in the second sense are presented. In this article, we always denote by the set of real numbers. Let X be anon-empty set and let , be a algebra of subsets of .
In reference , a mappingis a non-negative set function, and it is called a non-additive measure if it satisfies the following properties:
For any,and, imply.
For any,,and, imply.
The tripleis called a fuzzy measure space. Ifis a non-negative real-valued function, for any, denote, then it is easily shown that if, then .
In references [1,33], letbe a fuzzy measure space, letbe a non-negative measurable function, and; thus, the Sugeno integral (or the fuzzy integral) ofon setis defined as
Hereanddenote the operations sup and inf on, respectively.
In reference , letbe a fuzzy measure space,, and letbe non-negative measurable functions defined on. Thus,
For any constant, then;
Ifon the set, then;
Let, thenis the distribution associated toon. According to (5) and (6) of Proposition 1, we have
Thus, from a numerical point of view, the Sugeno integral canbe calculated as the solution of the equation .
In reference , letbe a pair of real numbers. A functionis said to be (s,m)-convex in the second sense if
holds for alland.
Some interesting and important inequalities for (s,m)-convex functions in the second sense functions can be found in [28,29,30,31,32]. Note that, if (s,m) = (1,1), then one obtains the definition of an ordinary convex function. If (s,m) = (s,1), then one obtains the definition of an s-convex function in the second sense. Denote bythe set of all (s,m)-convex functions in the second sense.
Let, then the inequality
Obviously, Inequality (5) is true when .
When , we let
The derivative of is
Since , ; therefore, . Thus, is an increasing function, and we have , for all . That is,
Obviously, if , then Inequality (5) is true. If , let in (8); thus, we have
Thus, . This completes the proof.
According to Lemma 1, we can easily prove thatis a (s,1)-convex function in the second sense () forand.
Let, then the inequality
Let for . Obviously, is increasing on the interval [0,1/2] and decreasing on the interval [1/2,1]. Thus, , for all . This completes the proof.
3. Sandor Type Inequalities for the Sugeno Integral Based on the(s,m)-Convex in the Second Sense
Sandor Type inequality, established by Sandor in , provides estimates of the mean value of a nonnegative and convex function with the following inequality:
Unfortunately, the following example shows that the Sandor type inequality for the Sugeno integral based on (s,m)-convex functions in the second senseis not valid.
Considerand letbe the Lebesgue measure on. If we take the function, then, by Remark 3, we know that. Calculate the Sugeno integralas Remark 2, and we obtain the following:
On the other hand,
This proves that the Sandor type inequality is not satisfied for the Sugeno integral based on (s,m)-convex functions in the second sense.
In this section, we will establish new Sandor type inequalities for the Sugeno integral based on (s,m)-convex functions in the second sense.
Letandis a (s,m)-convex function in the second sense, such that, and letbe the Lebesgue measure on. Thus,
Where is the positive real solution of the equation
As for , we have
According to Lemma 2, we have
By (15) and the non-negative assumption of function , we have for all . Thus, by (3) of Proposition 1 and Definition 2, we have
In order to calculate the integral , we consider the distribution function associated to on [a,b] which is given by
When , then
Let . Thus,
A straightforward calculus gives us that the positive real solution of the above equation is equal to .
By (1) of Proposition 1 and (16), we obtain
This completes the proof.
Let s = 1, m = 1,andis a convex function, such that. Letbe the Lebesgue measure on. Thus, one obtains the following:
whereis the positive real solution of the equation
Equation (19) can be solved as
where. This result is one of the main results of Caballero and Sadarangani .
Letandis an(s,m)-convex function in the second sense, such that. Letbe the Lebesgue measure on. Thus,
whereis the positive real solution of the equation
Similar to the proof of Theorem 1, we consider the function,
In this case, the distribution function is associated to on [a,b], which is given by
Let . Thus,
By (1) of Proposition 1 and (16), we obtain
This completes the proof.
Let s = 1, m = 1, and, andis a convex function, such that. Letbe the Lebesgue measure on. Thus, one obtains the following
Whereis the positive real solution of the equation
Equation (29) can be written as
Thus, the positive solution of Equation (30) is
where. This result is another main result of Caballero and Sadarangani .
For the case, we haveaccording to Equation (15). Thus, by (2) of Proposition 1,
Moreover, for convex function , i.e., s = m = 1, we have , which is in agreement with Caballero and Sadarangani .
Letbe the Lebesgue measure on. If we consider the function, then, by Remark 3, we know thatfor. Here, we let, so. By Theorem 1, through solving the equation, we obtain. Thus,
A straightforward calculation shows that . This also implies that the inequality can get a good estimate of .
Letbe the Lebesgue measure on. If we consider the function, then we can see thatis a convex function. That is,where s = m = 1. Therefore,, and, by Theorem 2, through solving the equation, we obtain. Hence, we get the following estimate:
Letbe the Lebesgue measure on. If we consider the function, then we can see thatis a convex function. That is,where s = m = 1. Therefore,, and, by Remark 5, we obtain the following estimate:
Letbe the Lebesgue measure on. If we consider the function, then we can see thatis a convex function. That iswhere s = m = 1. Thus,. By Theorem 1, through solving the equation, we obtain.
In Example 11 of Li et al. , they regarded as a general (1/3, 0, 3)-convex function, and they then obtained the result . A straightforward calculation shows that . This implies that the Sandor type inequalities constructed in this paper can more accurately estimate the Sugeno integral with respect to (s,m)-convex functions in the second sense.
In this paper, we establish an upper approximation for the Sugeno fuzzy integral of (s,m)-convex functions in the second sense based on classical Sendor type inequality, which is a useful tool for estimating Sugeno integrals. The results obtained in Caballero and Sadarangani  are special cases of this paper. In many applications, assumptions about the convexity of functions have interesting qualitative implications in many fields such as economics, biology, industrial engineering, etc. Fuzzy integrals were introduced by Sugeno in the early 1970s; they have since attracted the attention of many scholars and have been applied in many areas. Thus, the study of the Sugeno fuzzy integral for (s,m)-convex functions in the second sense is an important and interesting topic. In the future, we will study other properties and inequalities related to this convex function under other fuzzy integrals, such as the Choquet integral and the seminormed fuzzy integral.
The authors are grateful to the referees for a very careful reading of the manuscript and the suggestions that lead to the improvement of the paper. This work was supported by the National Natural Science Foundation of China (No. 71661012; Foundation of Big Data Laboratory of Chengyi College of Jimei University, the Natural Science Foundation of Jiangxi Province (No. 2014BAB201009), the Science and Technology Research Project of Jiangxi Provincial Education Department (No. GJJ14449), and the Natural Science Foundation of JXUST (No. NSFJ2014-G38).
Haiping Ren developed the idea for the article, and developed and wrote the manuscript. Guofu Wang provided examples, and Laijun Luo assisted in proving the theorems. All authors wrote the paper and have read and approved the final manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
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