Sandor Type Inequalities for Sugeno Integral with respect to General ([alpha],m,r)-Convex Functions

Abstract

The concept for general (α,m,r)-convex functions, as a generalization of convex functions, is introduced. Then, Sandor type inequalities for the Sugeno integral based on this kind of function are established. Our work generalizes the previous results in the literature. Finally, some conclusions and problems for further investigations are given.

Academic Editor:Ismat Beg

School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received 6 September 2015; Revised 3 December 2015; Accepted 7 December 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

As a tool for modeling nondeterministic problems, fuzzy measures and fuzzy integrals introduced by Sugeno in [1] have been successfully applied to various fields. The fuzzy integrals provide a useful tool in engineering and social choice, where the aggregation of data is required. However, they are restricted for the special operators used in the construction of these integrals. Thus, many researchers have generalized the Sugeno integral by using some other operators to replace the special operator(s) [figure omitted; refer to PDF] and/or [figure omitted; refer to PDF] . They obtained Choquet-like integral [2], Shilkret integral [3], [figure omitted; refer to PDF] -integral [4], generalized fuzzy integral [5], Sugeno-like integral [6], [figure omitted; refer to PDF] -generalized Sugeno integral [7], pseudointegral [8], interval-valued generalized fuzzy integral [9], and set-valued pseudointegral [10]. Garcia and Álvarez [11] presented two families of fuzzy integrals, the so-called seminormed fuzzy integrals and semiconormed fuzzy integrals. Klement et al. [12] provided a concept of universal integrals generalizing both the Choquet integral and the Sugeno integral. Wang and Klir [13] provided a general overview on fuzzy measurement and fuzzy integration.

The integral inequalities are significant mathematical tools both in theory and in application. Different integral inequalities including Chebyshev inequality, Jensen inequality, Hölder inequality, and Minkowski inequality are widely used in various fields of mathematics, such as probability theory, differential equations, decision-making under risk, forecasting of time series, and information sciences.

Convexity is one of the most powerful tools in establishing analytic inequalities. In particular, there are many important applications in the theory of higher transcendental functions. However, for many problems encountered in economics and engineering, convexity is unsuitable. Hence, it is natural to generalize convexity. Hanson [14] gave the notion of invexity as significant generalization of classical convexity. Ben-Israel and Mond [15] studied the preinvex functions, which are a special case of invex functions. Breckner [16] introduced the [figure omitted; refer to PDF] -convex functions and Varosanec [17] presented the [figure omitted; refer to PDF] -convex functions as a generalization of convex functions. In [18], Mihesan proposed the definition of [figure omitted; refer to PDF] -convex functions. For recent results and generalizations concerning [figure omitted; refer to PDF] -convex and [figure omitted; refer to PDF] -convex functions, see [19, 20]. Latif and Shoaib [21] discussed [figure omitted; refer to PDF] -preinvex functions and [figure omitted; refer to PDF] -preinvex functions. Gill et al. [22] provided the concept of [figure omitted; refer to PDF] -mean convex functions.

On the other hand, some scholars have shown that several integral inequalities hold both in the classical context and in the fuzzy context. Roman-Flores et al. investigated several kinds of fuzzy integral inequalities including Chebyshev type inequality [23], Young type inequality [24], Jensen type inequality [25], Hardy type inequality [26], Convolution type inequality [27], Stolarsky type inequality [28], and Markov type inequality [29]. Agahi et al. proved general Chebyshev type inequality [30], Hölder type inequality [31], Berwald type inequality [32], general Minkowski type inequality [33], and general Barnes-Godun-Levin type inequality [34] for the Sugeno integral. Caballero and Sadarangani presented Cauchy-Schwarz type inequality [35], Chebyshev type inequality [36], Fritz Carlson type inequality [37], and Sandor type inequality [38] for Sugeno integral. Mesiar and Ouyang proposed Chebyshev type inequality [39], Yong type inequality [40], general Chebyshev type inequality [41], and Minkowski type inequality [42] for Sugeno integral.

Caballero and Sadarangani [38] illustrated a Sandor type inequality of fuzzy integrals for convex function. The purpose of this paper is to investigate Sandor type inequalities for Sugeno integral related to the general [figure omitted; refer to PDF] -convexity, which generalize the previous results in the literature. Some examples are given to illustrate the results.

After some preliminaries and summarization of some previous known results in Section 2, Section 3 deals with general Sandor inequalities for Sugeno integral based on general [figure omitted; refer to PDF] -convex functions and some examples are given to illustrate the results. Section 4 focuses on Sandor inequalities for Sugeno integral based on general [figure omitted; refer to PDF] -concave functions. Finally, our results are applied to some special cases.

2. Preliminaries

In this section, we recall some basic definitions and properties of the fuzzy integral and introduce the notion of general [figure omitted; refer to PDF] -convex functions. For details, we refer the reader to [1, 13].

Definition 1 (see [18]).

Let [figure omitted; refer to PDF] be an interval, [figure omitted; refer to PDF] . A function [figure omitted; refer to PDF] is said to be convex on [figure omitted; refer to PDF] if [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . If the above inequalities reverse, then we say that the function [figure omitted; refer to PDF] is concave on [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be a nonempty set and let [figure omitted; refer to PDF] be the class of all subsets of [figure omitted; refer to PDF] .

Definition 2 (see [13]).

Let [figure omitted; refer to PDF] -algebra [figure omitted; refer to PDF] be a nonempty subclass of [figure omitted; refer to PDF] with the following properties:

(1) [figure omitted; refer to PDF] .

(2) If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] .

(3) If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be a fuzzy measure space, where [figure omitted; refer to PDF] is a nonempty set. Let [figure omitted; refer to PDF] be a [figure omitted; refer to PDF] -algebra of subsets of [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be a nonnegative, extended real-valued set function. We say that [figure omitted; refer to PDF] is a fuzzy measure if it satisfies the following:

(1) [figure omitted; refer to PDF] .

(2) [figure omitted; refer to PDF] and [figure omitted; refer to PDF] imply [figure omitted; refer to PDF] .

(3) [figure omitted; refer to PDF] , imply [figure omitted; refer to PDF] .

(4) [figure omitted; refer to PDF] , imply [figure omitted; refer to PDF] .

If [figure omitted; refer to PDF] is a nonnegative real-valued function defined on [figure omitted; refer to PDF] , we denote the set [figure omitted; refer to PDF] by [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Note that if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be a fuzzy measure space; we denote by [figure omitted; refer to PDF] the set of all nonnegative measurable functions with respect to [figure omitted; refer to PDF] .

Definition 3 (Sugeno [1]).

Let [figure omitted; refer to PDF] be a fuzzy measure space, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; the Sugeno integral (or the fuzzy integral) of [figure omitted; refer to PDF] on [figure omitted; refer to PDF] , with respect to the fuzzy measure [figure omitted; refer to PDF] , is defined as [figure omitted; refer to PDF] when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote the operations sup and inf on [figure omitted; refer to PDF] , respectively.

The properties of fuzzy integral are well known and can be found in [13].

Proposition 4.

Let [figure omitted; refer to PDF] be a fuzzy measure space, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; then one obtains the following:

(1) [figure omitted; refer to PDF] .

(2) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for nonnegative constant.

(3) [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] .

(4) [figure omitted; refer to PDF] .

(5) [figure omitted; refer to PDF] .

(6) [figure omitted; refer to PDF] there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

(7) [figure omitted; refer to PDF] there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Remark 5.

Consider the distribution function [figure omitted; refer to PDF] associated with [figure omitted; refer to PDF] on [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] . Then, due to (4) and (5) of Proposition 4, we have [figure omitted; refer to PDF] . Thus, from a numerical point of view, the fuzzy integral can be calculated solving the equation [figure omitted; refer to PDF] .

Definition 6.

Let [figure omitted; refer to PDF] be an interval, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be a continuous and monotonous function on [figure omitted; refer to PDF] . A function [figure omitted; refer to PDF] is said to be general [figure omitted; refer to PDF] -convex on [figure omitted; refer to PDF] if [figure omitted; refer to PDF] or [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . If the above inequalities reverse, then we say that the function [figure omitted; refer to PDF] is a general [figure omitted; refer to PDF] -concave function on [figure omitted; refer to PDF] .

Remark 7.

If, in Definition 6, [figure omitted; refer to PDF] (i.e., [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] ), then one obtains the definition of [figure omitted; refer to PDF] -convexity.

If, in Definition 6, [figure omitted; refer to PDF] , then one obtains the definition of general [figure omitted; refer to PDF] -mean convexity.

If, in Definition 6, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then one obtains the definition of [figure omitted; refer to PDF] -mean convexity [43].

If in Definition 6, [figure omitted; refer to PDF] , then one obtain the definition of general [figure omitted; refer to PDF] -convexity.

If, in Definition 6, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then one obtains the definition of [figure omitted; refer to PDF] -convexity [18].

If [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in Definition 6, one obtains the following classes of functions: increasing, [figure omitted; refer to PDF] -starshaped, starshaped, [figure omitted; refer to PDF] -convex, convex, and [figure omitted; refer to PDF] -convex, respectively.

3. Sandor Type Inequalities for Fuzzy Integrals Based on General [figure omitted; refer to PDF] -Convex Functions

The classical Sandor type inequality provides estimates of the mean value of a nonnegative and convex function: [figure omitted; refer to PDF] ; then, [figure omitted; refer to PDF] Unfortunately, the following example shows that Sandor type inequality for fuzzy integral based on general [figure omitted; refer to PDF] -convex functions is not valid.

Example 8.

Consider [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . If we take the function [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is a general [figure omitted; refer to PDF] -convex function. In fact, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Straightforward calculus shows that [figure omitted; refer to PDF] On the other hand, [figure omitted; refer to PDF]

This proves that the Sandor type inequality is not satisfied for Sugeno integral based on general [figure omitted; refer to PDF] -convex functions.

In this section, we will show general Sandor type inequalities for the Sugeno integral based on general [figure omitted; refer to PDF] -convex functions.

Theorem 9.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a continuous and monotonous function, let [figure omitted; refer to PDF] be a general [figure omitted; refer to PDF] -convex function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one has the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Proof.

As [figure omitted; refer to PDF] is a general [figure omitted; refer to PDF] -convex function for [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] By [figure omitted; refer to PDF] of Proposition 4, we have [figure omitted; refer to PDF] In order to calculate the right hand side of the last inequality, we consider the distribution function [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] and the solution of the equation [figure omitted; refer to PDF] . By [figure omitted; refer to PDF] of Proposition 4 and Remark 5, we get the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF] This completes the proof.

Remark 10.

If [figure omitted; refer to PDF] in Theorem 9, then [figure omitted; refer to PDF]

Example 11.

Consider [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . If we take the function [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is a general [figure omitted; refer to PDF] -convex function. In fact, [figure omitted; refer to PDF] By Theorem 9, we have [figure omitted; refer to PDF]

Now, we will prove the general case of Theorem 9.

Theorem 12.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a continuous and monotonous function, let [figure omitted; refer to PDF] be a general [figure omitted; refer to PDF] -convex function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Proof.

As [figure omitted; refer to PDF] is a general [figure omitted; refer to PDF] -convex function for [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] By [figure omitted; refer to PDF] of Proposition 4, we have [figure omitted; refer to PDF]

In order to calculate the right hand side of the last inequality, we consider the distribution function [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] and the solution of the equation [figure omitted; refer to PDF] . By [figure omitted; refer to PDF] of Proposition 4 and Remark 5, we get the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF] This completes the proof.

Remark 13.

If [figure omitted; refer to PDF] in Theorem 12, then [figure omitted; refer to PDF]

Example 14.

Consider [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . If we take the function [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is a general [figure omitted; refer to PDF] -convex function. In fact, [figure omitted; refer to PDF] By Theorem 12, we have [figure omitted; refer to PDF]

Now we consider some special cases of general [figure omitted; refer to PDF] -convex functions in Theorem 12.

Remark 15.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be an [figure omitted; refer to PDF] -convex function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Remark 16.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a continuous and monotonous function, let [figure omitted; refer to PDF] be a general [figure omitted; refer to PDF] -mean convex function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Remark 17.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be an [figure omitted; refer to PDF] -mean convex function, and [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Remark 18.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a continuous and monotonous function, let [figure omitted; refer to PDF] be a general [figure omitted; refer to PDF] -convex function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Remark 19.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in Theorem 12; then we obtain the general Sandor inequalities for fuzzy integral based on [figure omitted; refer to PDF] -convex functions.

Remark 20.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in Theorem 12; then we obtain the Sandor inequalities for fuzzy integral of convex functions [38].

4. Sandor Type Inequalities for Fuzzy Integral Based on General [figure omitted; refer to PDF] -Concave Functions

In this section, we provide Sandor type inequalities for fuzzy integral based on general [figure omitted; refer to PDF] -concave functions.

Theorem 21.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a continuous and monotonous function, let [figure omitted; refer to PDF] be a general [figure omitted; refer to PDF] -concave function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Proof.

As [figure omitted; refer to PDF] is a general [figure omitted; refer to PDF] -concave function for [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] By [figure omitted; refer to PDF] of Proposition 4, we have [figure omitted; refer to PDF] In order to calculate the right hand side of the last inequality, we consider the distribution function [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] and the solution of the equation [figure omitted; refer to PDF] . By [figure omitted; refer to PDF] of Proposition 4 and Remark 5, we get the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF] This completes the proof.

Remark 22.

If [figure omitted; refer to PDF] in Theorem 21, then [figure omitted; refer to PDF]

Example 23.

Consider [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . If we take the function [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is a general [figure omitted; refer to PDF] -concave function. In fact, [figure omitted; refer to PDF] By Theorem 21, we have [figure omitted; refer to PDF]

Now, we will prove the general case of Theorem 21.

Theorem 24.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a continuous and monotonous function, let [figure omitted; refer to PDF] be a general [figure omitted; refer to PDF] -concave function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Proof.

As [figure omitted; refer to PDF] is a general [figure omitted; refer to PDF] -concave function for [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] By [figure omitted; refer to PDF] of Proposition 4, we have [figure omitted; refer to PDF]

In order to calculate the right hand side of the last inequality, we consider the distribution function [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] and the solution of the equation [figure omitted; refer to PDF] . By [figure omitted; refer to PDF] of Proposition 4 and Remark 5, we get the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF] This completes the proof.

Remark 25.

If [figure omitted; refer to PDF] in Theorem 24, then [figure omitted; refer to PDF]

Example 26.

Consider [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . If we take the function [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is a general [figure omitted; refer to PDF] -concave function. In fact, [figure omitted; refer to PDF] By Theorem 24, we have [figure omitted; refer to PDF]

Now we consider some special cases of general [figure omitted; refer to PDF] -concave functions in Theorem 24.

Remark 27.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be an [figure omitted; refer to PDF] -concave function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Remark 28.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a continuous and monotonous function, let [figure omitted; refer to PDF] be a general [figure omitted; refer to PDF] -mean concave function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Remark 29.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be an [figure omitted; refer to PDF] -mean concave function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Remark 30.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a continuous and monotonous function, let [figure omitted; refer to PDF] be a general [figure omitted; refer to PDF] -concave function, and let [figure omitted; refer to PDF] be the Lebesgue measure on [figure omitted; refer to PDF] . Then, one obtains the following.

Case 1. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Case 2. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Case 3. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] satisfies the following equation: [figure omitted; refer to PDF]

Remark 31.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in Theorem 24; then we obtain the general Sandor type inequalities for fuzzy integral based on [figure omitted; refer to PDF] -concave functions.

Remark 32.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in Theorem 24; then we obtain the Sandor type inequalities for fuzzy integral of concave functions [38].

5. Conclusion

In this paper, we have proved the Sandor type inequalities for the Sugeno integral based on general [figure omitted; refer to PDF] -convex functions and general [figure omitted; refer to PDF] -concave functions. As open problems for future research, it would be interesting to investigate Sandor type inequalities for the Sugeno integral based on general [figure omitted; refer to PDF] -convex functions with respect to arbitrary fuzzy measure and explore Sandor type inequality to other generalizations of fuzzy integrals. We will investigate these problems in the future.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61273143, 61472424) and Fundamental Research Funds for the Central Universities (2013RC10, 2013RC12, and 2014YC07).

Conflict of Interests

The authors declare that they have no conflict of interests.

References

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