Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations

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, ((S) ∫ 0 f(x)pdμ)1/p((S) ∫ 0 g(x)qdμ)1/q ≤ (p−q/p − p−q + 1) ∨ (q−p/q − q−p + 1)(S) ∫ 0 f(x)g(x)dμ, where 1 < p < ∞, 1/p + 1/q = 1 and μ is the Lebesgue measure on R, holds if f and g 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.


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][7][8][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 [0, 1].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 1 <  < ∞, 1/ + 1/ = 1 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 2 Advances in Fuzzy Systems convex functions.Some examples are provided to illustrate the validity of the proposed inequalities.Definition 1.Let Σ be -algebra of subsets of R and let  : Σ → [0, ∞] be nonnegative, extended real-valued set function.We say that  is a fuzzy measure if and only if If  is a nonnegative real-valued function defined on R, then we denote by   = { ∈ R | () ≥ } = { ≥ } the level of , for  > 0, and We note that If  is a fuzzy measure on  ⊂ R, then we define the following: Definition 2. Let  be a fuzzy measure on (R, Σ).If  ∈ F  (R) and  ∈ Σ; then the Sugeno integral (or the fuzzy integral) of  on , with respect to the fuzzy measure , is defined as In particular, if  = R, 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 R and ,  ∈ F  (R), then Theorem 4 (see [13]).Let  : [0, ∞) → [0, ∞) be continuous and nonincreasing or nondecreasing functions and  be the Lebesgue measure on R. Let () ∫  0 () = .If 0 <  < , then () =  and ( − ) = , respectively.

Hölder Type Inequalities
The classical Hölder inequality in probability theory provides the following inequality [22]: where 1/+1/ = 1 or ∞ >  > 1 and ,  : [0, 1] → [0, ∞) 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.

Advances in Fuzzy Systems
Because  > 0 is arbitrary, we have and  * ≤  is trivial, which completes the proof.
The next example shows that the constant  , is optimal.
Then some straightforward calculus shows that and As  → ∞, we obtain that inequality (48) holds.