A Liapunov type inequality for Sugeno integrals

The classical Liapunov inequality shows an interesting upper bound for the Lebesgue integral of the product of two functions. This paper proposes a Liapunov type inequality for Sugeno integrals. That is, we show that holds for some constant H_s,t,r where 0<t<s<r,f:[0,1]→[0,∞) is a non-increasing concave function, and μ is the Lebesgue measure on R. We also present two interesting classes of functions for which the classical Liapunov type inequality for Sugeno integrals with H_s,t,r=1 holds. Some examples are provided to illustrate the validity of the proposed inequality.     

A Cauchy-Schwarz type inequality for fuzzy integrals

In this paper we prove a Cauchy-Schwarz type inequality for fuzzy integrals.     

A Chebyshev type inequality for Sugeno integral and comonotonicity

We supply a characterization of comonotonicity property by a Chebyshev type inequality for Sugeno integral.     

A sharp Hardy-type inequality of Sugeno integrals

This paper improves on previous work presenting a Hardy-type inequality for Sugeno integrals. Indeed, we show that for (S)_∫Df(x)dx?1,

     

New general extensions of Chebyshev type inequalities for Sugeno integrals

We provide new frameworks of Chebyshev type inequalities for Sugeno integrals on abstract spaces.     

Chebyshev type inequality for Choquet integral and comonotonicity

We supply a Chebyshev type inequality for Choquet integral and link this inequality with comonotonicity.     

A note on the Chebyshev-type inequality of Sugeno integrals

This paper improves on previous work presenting a Chebyshev-type inequality for Sugeno integrals. It derives a new inequality applicable to the Sugeno integrals and of real functions f,g on [0, 1]. Examples are given to illustrate the results.     

On the Chebyshev type inequality for seminormed fuzzy integral

The Chebyshev type inequality for seminormed fuzzy integral is discussed. The main results of this paper generalize some previous results obtained by the authors. We also investigate the properties of semiconormed fuzzy integral, and a related inequality for this type of integral is obtained.     

General Hardy type inequality for seminormed fuzzy integrals

Hardy type inequality for seminormed fuzzy integrals based on an aggregation function is studied in a rather general form. The main results of this paper generalize some previous results. Also, some conclusions are drawn and some problems for further investigations are given.     

Hermite–Hadamard inequality for fuzzy integrals

In this paper we prove a Hermite–Hadamard type inequality for fuzzy integrals. Some examples are given to illustrate the results.     

Real-valued Choquet integrals for set-valued mappings

In this paper a new kind of real-valued Choquet integrals for set-valued mappings is introduced, and some elementary properties of this kind of Choquet integrals are studied. Convergence theorems of a sequence of Choquet integrals for set-valued mappings are shown. However, in the case of the monotone convergence theorem of the nonincreasing sequence of Choquet integrals for set-valued mappings, we point out that the integrands must be closed. Specially, this kind of real-valued Choquet integrals for set-valued mappings can be regarded as the Choquet integrals for single-valued functions.     

A note on a Carlson-type inequality for the Sugeno integral

In this short note, we present a general version of Carlson’s inequality for the Sugeno integral, which generalizes some recent results obtained by others.     

Berwald type inequality for Sugeno integral

Nonadditive measure is a generalization of additive probability measure. Sugeno integral is a useful tool in several theoretical and applied statistics which has been built on non-additive measure. Integral inequalities play important roles in classical probability and measure theory. The classical Berwald integral inequality is one of the famous inequalities. This inequality turns out to have interesting applications in information theory. In this paper, Berwald type inequality for the Sugeno integral based on a concave function is studied. Several examples are given to illustrate the validity of this inequality. Finally, a conclusion is drawn and a problem for further investigations is given.     

Sandor’s inequality for Sugeno integrals

In this paper we prove a fuzzy integral inequality for convex functions. Our results improve recent results that appear in literature. Some examples are given to illustrate our theorems.     

Fuzzy measure of fuzzy events defined by fuzzy integrals

We define the concept of fuzzy measure of a fuzzy event by using a general form of fuzzy integral proposed by Murofushi, called fuzzy t-conorm integral, encompassing previous definitions. Zadeh defined the probability measure of a fuzzy event, and later the possibility measure of fuzzy event. Using a duality property of fuzzy t-conorm integral, we propose a general definition of fuzzy measure of fuzzy events, which is compatible with previous definitions of Zadeh, and possesses all properties of a fuzzy measure, in particular the duality property. Using our definition, we examine the case of decomposable measures and belief functions. A comparison with previous works is provided.     

Gaussian quadratures for approximate of fuzzy integrals

In this paper the integration formulas of Gaussian methods with positive coefficient for fuzzy integrations are discussed and then are followed by convergence theorem. The proposed algorithms are illustrated and compared with Newton Cot’s methods [T. Allahviranloo, Newton Cot’s methods for integration of fuzzy functions, Appl. Math. Comp., in press] by solving some numerical examples.     

On the level-continuity of fuzzy integrals

In this paper we define the level-convergence of measurable functions on a fuzzy measure space, by using the closure operator in the Moore sense. We study some of the properties of this convergence and give conditions for the continuity of the fuzzy integral in relation to the level-convergence.     

A Cauchy-Schwarz type inequality for bilinear integrals on positive measures

If is Borel measurable, define for -finite positive Borel measures on the bilinear integral expression

We give conditions on such that there is a constant , independent of and , with

Our results apply to a much larger class of functions than known before.

     

On the monotone convergence theorem for conormed-seminormed fuzzy integral

[In this paper, under the condition that the t-conorm and t-seminorm are continuous, we prove the monotone convergence theorem of conormed-seminormed fuzzy integral for non-decreasing monotone measurable function sequence; also we show that the result cannot be extended to the case of the non-increasing function sequence.