# General Fritz Carlson's Type Inequality for Sugeno Integrals

- Xiaojing Wang
^{1}and - Chuanzhi Bai
^{1}Email author

**2011**:761430

https://doi.org/10.1155/2011/761430

© X. Wang and C. Bai. 2011

**Received: **18 August 2010

**Accepted: **20 January 2011

**Published: **7 February 2011

## Abstract

Fritz Carlson's type inequality for fuzzy integrals is studied in a rather general form. The main results of this paper generalize some previous results.

## 1. Introduction and Preliminaries

Recently, the study of fuzzy integral inequalities has gained much attention. The most popular method is using the Sugeno integral [1]. The study of inequalities for Sugeno integral was initiated by Román-Flores et al. [2, 3] and then followed by the others [4–11].

Now, we introduce some basic notation and properties. For details, we refer the reader to [1, 12].

Suppose that is a -algebra of subsets of , and let be a nonnegative, extended real-valued set function. We say that is a fuzzy measure if it satisfies

(3) , imply (continuity from below),

(4) , , , imply (continuity from above).

If is a nonnegative real-valued function defined on , we will denote by the -level of for , and is the support of . Note that if , then .

Let be a fuzzy measure space; by we denote the set of all nonnegative -measurable functions with respect to .

Definition 1.1 (see [1]).

where and denote the operations and on , respectively.

does not hold.

The following properties of the fuzzy integral are well known and can be found in [12].

Proposition 1.2.

Let be a fuzzy measure space, with and ; then

(2) , for a nonnegative constant,

Remark 1.3.

Thus, from a numerical point of view, the Sugeno integral can be calculated by solving the equation .

Recently, Caballero and Sadarangani [8] have shown that in general, the Carlson's integral inequality is not valid in the fuzzy context. And they presented a fuzzy version of Fritz Carlson's integral inequality as follows.

Theorem 1.4.

In this paper, our purpose is to give a generalization of the above Fritz Carlson's inequality for fuzzy integrals. Moreover, we will give many interesting corollaries of our main results.

## 2. Main Results

This section provides a generalization of Fritz Carlson's type inequality for Sugeno integrals. Before stating our main results, we need the following lemmas.

Lemma 2.1 (see [11]).

If the fuzzy measure in Lemma 2.1 is the Lebesgue measure, then is satisfied readily. Thus, by Lemma 2.1, we have the following.

Corollary 2.2 (see [8]).

Definition 2.3.

An important property of comonotone functions is that for any real numbers , , either or .

Note that two monotone functions (in the same sense) are comonotone.

Theorem 2.4.

Proof.

Since the values of are arbitrary, we obtain the desired inequality. Similarly, for the case we can get the desired inequality too.

From Theorem 2.4, we get the following.

Corollary 2.5 (see [15]).

holds.

If the fuzzy measure in Corollary 2.5 is the Lebesgue measure and , then and are satisfied readily. Thus, by Corollary 2.5, we obtain

Corollary 2.6 (see [2]).

holds.

The following result presents a fuzzy version of generalized Carlson's inequality.

Theorem 2.7.

Proof.

This inequality implies that (2.11) holds

By Theorem 2.7, we have the following.

Corollary 2.8.

Theorem 2.9.

Proof.

which implies that (2.17) holds. Similarly, we can obtain (2.17) by of [7, Theorem 2] for the case of .

From Theorem 2.9 and Corollary 2.8, we have the following.

Theorem 2.10.

Theorem 2.11.

where and are as in (2.21) and (2.23), respectively.

If , and , and , , and , respectively, then Corollary 2.8 reduces to Theorem 1.4, and the following Corollaries 2.12 and 2.13.

Corollary 2.12.

Corollary 2.13.

Remark 2.14.

Corollary 2.8 is a generalization of the main result in [8, Theorem 1].

If , , then Corollary 2.8 reduces to the following corollary.

Corollary 2.15.

Consider on . This function is nonincreasing ( ), nonnegative and convex ( ).

Thus, by Theorem 2.11 we can get the following corollary.

Corollary 2.16.

Thus, by Theorem 2.11 (set ) we can get the following corollary.

Corollary 2.17.

Thus, by Theorem 2.10 (set , , , ) we can get the following corollary.

Corollary 2.18.

## Declarations

### Acknowledgments

The authors would like to thank the referees for reading this work carefully, providing valuable suggestions and comments. This work is supported by the National Natural Science Foundation of China (no. 10771212).

## Authors’ Affiliations

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## Copyright

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