General Fritz Carlson's Type Inequality for Sugeno Integrals

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

(1),

(2) and imply (monotonicity);

(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]).

Let be a fuzzy measure space, with , and , then the Sugeno integral (or fuzzy integral) of on with respect to the fuzzy measure is defined by

(1.1)

where and denote the operations and on , respectively.

It is well known that the Sugeno integral is a type of nonlinear integral; that is, for general cases,

(1.2)

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

(1),

(2), for a nonnegative constant,

(3)if on then ,

(4)if then ,

(5),

(6),

(7) there exists such that ,

(8) there exists such that .

Remark 1.3.

Let be the distribution function associated with on , that is, . By (5) and (6) of Proposition 1.2

(1.3)

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

Fritz Carlson's integral inequality states [13, 14] that

(1.4)

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.

Let be a nondecreasing function and the Lebesgue measure on . Then,

(1.5)

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]).

Let be a fuzzy measure space, , , , and . Then

(2.1)

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]).

Let be a -measurable function with the Lebesgue measure and . Then

(2.2)

Definition 2.3.

Two functions are said to be comonotone if for all ,

(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.

Let be a fuzzy measure space, and and comonotone functions, with , and . Then

(2.4)

Proof.

If or then the inequality is obvious. Now choose , such that

(2.5)

Then by (8) of Proposition 1.2, there exist and such that

(2.6)

As and are comonotone functions, then either or . Suppose that . In this case, we have the following:

(2.7)

Therefore, by applying (8) of Proposition 1.2 again, we find that

(2.8)

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]).

Let be an arbitrary fuzzy measure on and be two real-valued measurable functions such that and . If and are increasing (or decreasing) functions, then the inequality

(2.9)

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]).

Let be two real-valued functions, and let be the Lebesgue measure on . If are both continuous and strictly increasing (decreasing) functions, then the inequality

(2.10)

holds.

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

Theorem 2.7.

Let be a fuzzy measure space, , and , and and are comonotone functions, respectively, with , , , , and . Then

(2.11)

where .

Proof.

By Lemma 2.1, for , we have the following:

(2.12)

Multiplying these inequalities, we get that

(2.13)

By Theorem 2.4

(2.14)

Substitutes (2.14) into (2.13), we obtain

(2.15)

This inequality implies that (2.11) holds

By Theorem 2.7, we have the following.

Corollary 2.8.

Assume that . Let are increasing (or decreasing) functions and the Lebesgue measure on . Then be

(2.16)

where .

Theorem 2.9.

Let be a -measurable function with the Lebesgue measure. If () is a convex function such that, , then

(2.17)

Proof.

Firstly, we consider the case of . As is a convex function, we have by Theorem  1 of Caballero and Sadarangani [7] that

(2.18)

By Corollary 2.2 and (2.18), we get

(2.19)

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.

Assume that . Let be increasing (or decreasing) functions and the Lebesgue measure on . If ( ) or ( ) is a convex function such that or , then

(2.20)

where

(2.21)

or

(2.22)

where

(2.23)

Theorem 2.11.

Assume that . Let be increasing (or decreasing) functions and the Lebesgue measure on . If ( )and ( ) are two convex functions such that and , then,

(2.24)

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

Straightforward calculus shows that

(2.25)

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.

Let be a nondecreasing function and the Lebesgue measure on . Then,

(2.26)

Corollary 2.13.

Let be a nondecreasing function and the Lebesgue measure on . Then,

(2.27)

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.

Let be a nondecreasing function and the Lebesgue measure on . Then

(2.28)

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

Let , , and . As and , we have the following

(2.29)

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

Corollary 2.16.

Let be a nonincreasing function and the Lebesgue measure on . Then,

(2.30)

Consider and on . Obviously, and are nonnegative, nondecreasing and convex on the interval . Let , then, we have the following:

(2.31)

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

Corollary 2.17.

Let be a nondecreasing function and the Lebesgue measure on . Then,

(2.32)

Consider on . Obviously, this function is nonnegative, nondecreasing (), and nonconvex (). But is convex. Set , then we obtain

(2.33)

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

Corollary 2.18.

Let be a nondecreasing function and the Lebesgue measure on . Then

(2.34)