Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations

Abstract

The classical Hölder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes Hölder type inequalities and reverse Hölder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions. One of the interesting results is that the inequality, where and is the Lebesgue measure on holds if and are nonincreasing and concave functions. As a special case, we consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or convex functions. Some examples are provided to illustrate the validity of the proposed inequalities.

1. Introduction and Preliminaries

A number of studies have examined the Sugeno integral since its introduction in 1974 [1]. Ralescu and Adams [2] generalized a range of fuzzy measures and provided several equivalent definitions of fuzzy integrals. Wang and Klir [3] provided an overview of fuzzy measure theory.

Caballero and Sadarangani [4, 5] proved a Hermite-Hadamard type inequality and a Fritz Carlson’s inequality for fuzzy integrals. Román-Flores et al. [6–9] presented several new types of inequalities for Sugeno integrals, including a Hardy type inequality, a Jensen type inequality, and some convolution type inequalities. Flores-Franulič et al. [10, 11] presented Chebyshev’s inequality and Stolarsky’s inequality for fuzzy integrals. Mesiar and Ouyang [12] generalized Chebyshev type inequalities for Sugeno integrals. Ouyang and Fang [13] generalized their main results to prove some optimal upper bounds for the Sugeno integral of the monotone function in [8]. Ouyang et al. [14] generalized a Chebyshev type inequality for the fuzzy integral of monotone functions based on an arbitrary fuzzy measure. Hong [15] extended previous research by presenting a Hardy-type inequality for Sugeno integrals in [6]. Hong [16] proposed a Liapunov type inequality for Sugeno integrals and presented two interesting classes of functions for which the classical Liapunov type inequality for Sugeno integrals holds. In Hong et al. [17] we consider Steffensen’s integral inequality for the Sugeno integral where is a nonincreasing and convex function and is a nonincreasing function defined on Hong [18] proposed a Berwald type inequality and a Favard type inequality for Sugeno integrals. Many researchers [19, 20] have also studied the inequalities for other fuzzy integrals.

Recently, Wu et al. [21] considered Hölder type inequalities for Sugeno integrals. However, they did not examine their results under usual multiplication operations and did not make the essential assumption of for the classical Hölder inequality.

In this paper, we propose Hölder type inequalities for Sugeno integrals and find optimal constants for which these inequalities hold for nonincreasing concave or convex functions under usual multiplication operations. We also propose a reverse Hölder type inequality for Sugeno integrals. As a special case, we consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or convex functions. Some examples are provided to illustrate the validity of the proposed inequalities.

Definition 1. Let be -algebra of subsets of and let be nonnegative, extended real-valued set function. We say that is a fuzzy measure if and only if(a).(b) and imply (monotonicity)(c) and imply (continuity form below)(d), , and imply (continuity form above)

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

We note that

If is a fuzzy measure on , then we define the following:

Definition 2. Let be a fuzzy measure on . If and ; then the Sugeno integral (or the fuzzy integral) of on , with respect to the fuzzy measure , is defined asIn particular, if , then

The following properties of the Sugeno integral are well known and can be found in [1].

Proposition 3 (see [1]). If is a fuzzy measure on and , then(i)(ii) for any constant (iii), if on (iv)(v)(vi) there exists such that (vii) there exists such that

Theorem 4 (see [13]). Let be continuous and nonincreasing or nondecreasing functions and be the Lebesgue measure on . Let If , then and , respectively.

2. Hölder Type Inequalities

The classical Hölder inequality in probability theory provides the following inequality [22]:where or and are integrable functions. inequality (5) shows an interesting upper bound for the Lebesgue integral of the product of two functions. In general, inequality (5) does not hold for the Sugeno integral as demonstrated by the following example.

Example 5. Letfor Then, some straightforward calculus shows thatand because is nonincreasing and continuous, by Theorem 4,Consider thatLet and ThenTherefore, inequality (5) does not hold for Sugeno integrals.

In this context, this section presents Hölder type inequalities derived from (5) for Sugeno integrals. For this we first consider the following lemma.

Lemma 6. Let and be any nonincreasing concave functions on and . Then we have

Proof. LetThen, by Theorem 4, we haveWe consider the case of and that of to be similar. Note that because , we haveandBecause is nonincreasing and continuous, by the Intermediate Value Theorem and Theorem 4, there exists such thatIt is then easy to check that and that Therefore, should satisfy the equationthat is,

Lemma 7. Let and be any nonincreasing convex functions on and . Then we have

Proof. LetThen, by Theorem 4, we haveWe consider the case of and that of to be similar. Note that because , we haveandBecause is nonincreasing and continuous, by the Intermediate Value Theorem and Theorem 4, there exists such thatIt is then easy to check that Let for any Then , and we have, for any , Because is arbitrary, we haveand is trivial, which completes the proof.

Proposition 8. Suppose that and are non-increasing concave functions on and that is the Lebesgue measure on . Then for , there is no such that the Hölder type inequalityholds.

Proof. Let Then by Lemma 6, Letting and , thenand, similarly,Therefore, , which completes the proof.

Proposition 9. Suppose that and are nonincreasing convex functions on and that is the Lebesgue measure on . Then, for , there is no such that the Hölder type inequality holds.

Proof. Let Then, by Lemma 7, Let and Then,and, similarly,Therefore , which completes the proof.

Theorem 10 (Hölder type inequality for concave functions). Suppose that and are nonincreasing concave functions on such that and that is the Lebesgue measure on . Then, for , the inequality holds.

Proof. LetThen, by Lemma 6, Now, consider that, for that is, is increasing with respect to and decreasing with respect to Suppose that ; then , because is concave and constant on , and, hence, we haveThen, by letting , we haveand, similarly, we have, for ,Therefore, we have

Corollary 11 (Cauchy-Schwarz type inequality for concave functions). Suppose that and are nonincreasing concave functions on such that and that is the Lebesgue measure on . Then, for , the inequalityholds.

Example 12. Let be the usual Lebesgue measure on . If we take the functions defined by , then . Then some straightforward calculus shows thatandTherefore,andTherefore, inequality (48) holds.

The next example shows that the constant is optimal.

Example 13. Let be the usual Lebesgue measure on . If we take the functions defined by , then , and, hence, Then some straightforward calculus shows thatandTherefore,andAs , we obtain that inequality (48) holds.

Theorem 14 (Hölder type inequality for convex functions). Suppose that and are nonincreasing convex functions on such that and that is the Lebesgue measure on . Then, for , the inequality holds.

Proof. Let Then, by Lemma 7,Now, consider that, for ,Suppose that . Then , and, hence, we haveThen we haveSimilarly, we have, for ,Therefore, we have

Corollary 15 (Cauchy-Schwarz type inequality for convex functions). Suppose that and are nonincreasing convex functions on such that and that is the Lebesgue measure on . Then, for , the inequalityholds.

Example 16. Let be the usual Lebesgue measure on . If we take the functions defined by then . Then some straightforward calculus shows thatandTherefore,andThus, inequality (65) holds.

3. Reverse Hölder Type Inequalities

In this section, we consider a reverse Hölder type inequality derived from (5) for Sugeno integrals. For this, we first consider the following lemma.

Lemma 17. Let and be any nonincreasing concave functions on and . Then we have

Proof. LetThen, by Theorem 4, we haveWe consider the case of and that of to be similar. Note that because , we haveandBecause is nonincreasing and continuous, by the Intermediate Value Theorem and Theorem 4, there exists such thatIt is then easy to check that andTherefore, should satisfy the equationthat is,

Lemma 18. Let and be any nonincreasing convex functions on and . Then we have

Proof. LetThen, by Theorem 4, we haveWe consider the case of and that of to be similar. Note that since , we haveandBecause is nonincreasing and continuous, by the Intermediate Value Theorem and Theorem 4, there exists such thatIt is now easy to check thatLet for any Then , and we have, for any , Because is arbitrary, we haveand is trivial, which completes the proof.

Theorem 19 (reverse Hölder type inequality for concave function). Suppose that and are nonincreasing concave functions on and that is the Lebesgue measure on . Then for the inequality where holds.

Proof. Let Then, by Lemma 17,We first consider that, for ,Becausethen We also note thatThus, we haveSimilarly, we have, for ,Therefore, we havewhich completes the proof.

Corollary 20 (reverse Cauchy-Schwarz type inequality for concave functions.). Suppose that and are nonincreasing concave functions on and that is the Lebesgue measure on . Then the inequalityholds.

Example 21. Let be the usual Lebesgue measure on . If we take the functions defined by . Then some straightforward calculus shows thatandThenandTherefore, inequality (98) holds.

Proposition 22. Suppose that and are nonincreasing convex functions on and that is the Lebesgue measure on . Then, for , there is no such that the Hölder type inequality holds.

Proof. LetThen, by Lemma 18,Let and Thenand, similarly,Therefore, , which completes the proof.

Theorem 23 (reverse Hölder type inequality for convex functions). Suppose that and are nonincreasing convex functions on such that and that is the Lebesgue measure on . Then for the inequality holds.

Proof. Let Then, by Lemma 18, We first consider that, for , the function is decreasing with respect to and increasing with respect to . Suppose that . Then , and, hence, we haveThen we haveSimilarly, we have, for ,Therefore, we have

Corollary 24 (reverse Cauchy-Schwarz type inequality for convex functions). Suppose that and are nonincreasing convex functions on such that and that is the Lebesgue measure on . Then for , the inequalityholds.

Example 25. Let be the usual Lebesgue measure on . If we take the functions defined by , then . Then some straightforward calculus shows thatandTherefore,andTherefore, inequality (115) holds.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03027869).