A Chebyshev type inequality for fuzzy integrals

In this paper, we prove a Chebyshev type inequality for fuzzy integrals. More precisely, we show that:

where μ is the Lebesgue measure on and f,g:[0,1]→[0,∞) are two continuous and strictly monotone functions, both increasing or both decreasing. Also, some examples and applications are presented.     

Chebyshev type inequality for Choquet integral and comonotonicity

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

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.     

A Chebyshev type inequality for Sugeno integral and comonotonicity

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

Refinement of an inequality of P. L. Chebyshev

Let P _n denote the linear space of polynomials p(z:=Σ _k=0 ⁿ a _k (p)z ^k of degree ≦ n with complex coefficients and let |p|_[?1,1]: = max _x∈[?1,1]|p(x)| be the uniform norm of a polynomial p over the unit interval [?1, 1]. Let t _n ∈ P _n be the n ^th Chebyshev polynomial. The inequality $$ \frac{{\left| p \right|_{\left[ { - 1,1} \right]} }} {{\left| {a_n (p)} \right|}} \geqq \frac{{\left| {t_n } \right|_{\left[ { - 1,1} \right]} }} {{\left| {a_n (t_n )} \right|}},p \in P_n $$ due to P. L. Chebyshev can be considered as an extremal property of the Chebyshev polynomial t _n in P _n . The present note contains various extensions and improvements of the above inequality obtained by using complex analysis methods.     

Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality

We prove two theorems on upper and lower bounds for probabilities in the multidimensional case. We generalize and improve the Prokhorov multidimensional analog of the Chebyshev inequality and establish a multidimensional analog of the generalized Kolmogorov probability estimate. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 573–576, April, 2006.     

Stability of the linear inequality method for rational Chebyshev approximation

The linear inequality method is an algorithm for discrete Chebyshev approximation by generalized rationals. Stability of the method with respect to uniform convergence is studied. Analytically, the method appears superior to all others in reliability.     

Chebyshev type inequalities for pseudo-integrals

Chebyshev type inequalities for two classes of pseudo-integrals are shown. One of them concerning the pseudo-integrals based on a function reduces on the g-integral where pseudo-operations are defined by a monotone and continuous function g. Another one concerns the pseudo-integrals based on a semiring ([a,b],max,⊙), where ⊙ is generated. Moreover, a strengthened version of Chebyshev’s inequality for pseudo-integrals is proved.     

A bivariate Markov inequality for Chebyshev polynomials of the second kind

This note presents a Markov-type inequality for polynomials in two variables where the Chebyshev polynomials of the second kind in either one of the variables are extremal. We assume a bound on a polynomial at the set of even or odd Chebyshev nodes with the boundary nodes omitted and obtain bounds on its even or odd order directional derivatives in a critical direction. Previously, the author has given a corresponding inequality for Chebyshev polynomials of the first kind and has obtained the extension of V.A. Markov’s theorem to real normed linear spaces as an easy corollary.To prove our inequality we construct Lagrange polynomials for the new class of nodes we consider and give a corresponding Christoffel–Darboux formula. It is enough to determine the sign of the directional derivatives of the Lagrange polynomials.     

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that

A Noether type inequality

This paper gives a Noether type inequality of a minimal Gorenstein 3-fold of general type whose canonical map is generically finite.     

A Fisher type inequality

In this paper we study subsets of a finite set that intersect each other in at most one element. Each subset intersects most of the other subsets in exactly one element. The following theorem is one of our main conclusions. Let S₁,… S_m be m subsets of an n-set S with |S₁| ? 2 (l = 1, …,m) and |S_i ∩ S_j| ? 1 (i ≠ j; i, j = 1, …, m). Suppose further that for some fixed positive integer c each S_i has non-empty intersection with at least m ? c of the remaining subsets. Then there is a least positive integer M(c) depending only on c such that either m ? n or m ? M(c).     

Whittle type inequality for demisubmartingales

A Whittle type inequality for demisubmartingales is derived and a strong law of large numbers for functions of a demisubmartingale is obtained.

     

On a Littlewood-Paley type inequality

The following is proved: If is a function harmonic in the unit ball and if then the inequality

holds, where is the nontangential maximal function of This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well.