The Iyengar inequality,revisited

Detailed analysis shows that the famous Iyengar inequality actually says that the Trapezoidal formula is a central algorithm for approximating integrals over an appropriate interval for the class of functions whose derivatives are bounded by a positive number K in L _∞-sense. The inherent nonlinearity from central algorithms reflects the importance of the Iyengar inequality and thus makes familiar linear methods malfunction when one tries to generalize it. It is shown that the generalization depends on a nonlinear system of equations satisfied by a set of free nodes of a perfect spline. Explicit constructions are obtained in the spirit of the Iyengar inequality for the class of functions whose rth (r≤4) derivatives are bounded by a positive number K in L _∞-sense because a closed solution to the nonlinear system is only available for r≤4. Connections with computational mathematics, especially with best interpolation and best quadrature, are discussed. Numerical experiments are also included. AMS subject classification (2000)  65D30, 41A17     

On weighted Iyengar type inequalities on time scales

In this study, we establish some new weighted Iyengar type integral inequalities using Steffensen’s inequality on time scales.     

Note on a Littlewood-Paley inequality

We show that a recent result of Littlewood-Paley type, due to the author, is essentially best-possible.     

On an inequality of Friedrich's type

In this paper the dependence of the constant in the inequality on simply connected bounded domains is found.

     

Note on a Diophantine inequality in several variables

We establish estimates for the number of points that belong to an aligned box in in terms of certain exponential sums. These generalize previous results that were known only in case .

     

On an integral inequality of the Wirtinger type

In this paper, a new Picone-type identity for the quasilinear differential operator of the second order is used to establish an integral inequality involving functions and their derivatives which generalizes the classical Wirtinger inequality.     

On an inequality of Sidon type for trigonometric polynomials

We establish a lower bound for the uniform norm of the trigonometric polynomial of special form via the sum of the L ¹-norms of its summands. This result generalizes a theorem due to Kashin and Temlyakov, which, in turn, generalizes the classical Sidon inequality.     

A certain inequality for an entire function of exponential type

This research was supported by GrantN 93-011-197 from the Russian Foundation of Fundamental Research     

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.     

Fuzzy Chebyshev type inequality

A Chebyshev type inequality for Sugeno integral is shown. Previous results of Flores-Franulič and Román-Flores [A. Flores-Franulič, H. Román-Flores, A Chebyshev type inequality for fuzzy integrals, Applied Mathematics and Computation 190 (2007) 1178–1184] are generalized. Several illustrated examples are given. As an application, a fuzzy Stolarsky’s inequality is obtained.     

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).     

对一个不等式的一点注记

讨论了不等式bx+y-ax+y/bx-ax≥x+y/x(ab)r/2及其逆成立或不成立的一切情形,其中x,y∈R x≠0 a,b＞0,a≠b.