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With the development of fuzzy measure theory, the integral inequalities based on Sugeno integral are extensively investigated. We concern on the inequalities of Choquuet integral. The main purpose of this paper is to prove the H?lder inequality for any arbitrary fuzzy measure-based Choquet integral whenever any two of these integrated functions f, g and h are comonotone, and there are three weights. Then we prove Minkowski inequality and Lyapunov inequality for Choquet integral. Moreover, when any two of these integrated functions f
_{1}, f
_{2},
…, f
_{n} are comonotone, we also obtain the H
ölder inequality, Minkowski inequality and Lyapunov inequality hold for Choquet integral.

The Choquet integral, introduced in [

Ralescu and Adams [

but it is generally nonlinear with respect to its integral due to the nonadditivity of

So, in some sense, the Choquet integral ia a kind of fuzzy integral. But, unlike the Sugeno integral [

Integral inequalities are useful tools in several theoretical and applied fields. For more information on classical inequalities, we refer the reader to the distinguished monograph [

Section 2 consists of some preliminaries and notations about Choquet integral. In section 3, we prove the Hölder inequality for arbitrary fuzzy measure-based Choquet integral whenever any two of these integrated functions are comonotone. Then, we prove Minkowski inequalities and Lyapunov inequality for arbitrary fuzzy measure-based Choquet integral whenever any two of these integrated functions are comonotone. And including several examples. Finally, some conclusions are drawn.

In this section we recall some basic definitions and previous results that will be used in the sequel.

As usual we denote by R the set of real numbers. Let X be a nonempty set,

Definition 1. ( [

(FM1)

(FM2)

(FM3)

(FM4)

When

Definition 2. ( [

where

Since f in Definition 2 is measurable, we know that

The Choquet integral has some properties of the Lebesgue integral. These properties are listed in the following theorem.

Theorem 1. ( [

1)

2)

3)

4) If

5) If

6)

Unlike the Lebesgue integral, the Choquet integral is generally nonlinear with respect to its integrand due to the nonadditivity of

for some nonnegative measurable functions f and g. But when integrand f and g satisfying the properties of comonotone, then we have

This is the properties of Choquet integral of comonotone additivity. Then we give the definition of two functions comonotonicity.

Definition 3. ( [

Clearly, if f and g are comonotone, then for all nonnegative real numbers

This section is devoted to providing Hölder inequality for Choquet integral, when there are three integrand and three weights. And these integrand satisfying the properties of comonotone additivity. Then we prove Hölder inequality for Choquet integral about a finite number of integrands and finite weights appears as its corollary.

In this paper, we suppose any two of these nonnegative measurable functions

Theorem 2 (Hölder inequality). Let

measurable functions. When any two of f, g and h are comonotone, and

holds.

Proof. By Theorem 3.1 [

holds. Let

the product of a finite number of measurable functions still can be measurable, we have fg is nonnegative measurable function. And fg and h are comonotone for any

holds. Let

holds. Then, by the inequalities (4) and (5), we obtain

This completes the proof.

Then, let us review examples illustrating the previous result.

Example 1. Let

the class of all Borel sets in

In a similar manner, we calculate that

By the inequality

Then, we obtain

When the integrand

Example 2. Let

Then, we can calculate that

So by the inequality

where

From the above two examples we can get, f, g and h be nonnegative measurable functions, when any two of f, g and h are comonotone, and

Hölder inequality for Choquet integral about a finite number of integrands and finite weights appears in the following corollary.

Corollary 1. Let _{n} be nonnegative measurable

functions. When any two of

holds.

As the application of Hölder inequality for Choquet integral, we will prove Minkowski inequality. First, we prove the following lemma.

Lemma 1. Let f, g and

Proof. For any

If

Then, the inequality (9) holds.

The case that when

We have proved the inequality (9) holds, when

So, we obtain the functions

As so far, we prove any two of these functions

This completes the proof.

Then the Minkowski inequality for Choquet integral is given in the following theorem.

Theorem 3 (Minkowski inequality). Let

holds for any

Proof. When

Obviously, the inequality (10) holds.

When

In the same method, we get

Hence,

This completes the proof.

Example 3 Let

In the same way,we calculate that

Then, we get

If there is a finite nonnegative measurable function, the Minkowski inequality for Choquet integral holds or not. First, we have to prove the following corollary.

Corollary 2. Let

these functions

Corollary 3. Let

holds, for any

Theorem 4 (Lyapunov inequality). Let

equality,

holds.

Proof. Let

And by

Remark 1. Let

for any

Corollary 4. Let

In this paper, we prove the Hölder inequalities for any arbitrary fuzzy measure based on Choquet integral whenever any two of these integrated functions f, g and h are comonotone. As its application, we also prove Minkowski inequality and Lyapunov inequality for Choquet integral. Moreover, we also obtain whenever any two of these integrated functions

This work was supported by the National Natural Science Foundation of China (no. 51374199).

XiuliYang,XiaoqiuSong,LeileiHuang, (2015) Some General Inequalities for Choquet Integral. Applied Mathematics,06,2292-2299. doi: 10.4236/am.2015.614201