关于Hilbert积分不等式

本文通过引入一个适当的形如Ｆ（ｘ，ｙ）的正定二次型证明了Ｈｉｌｂｅｒｔ积分不等式可以得到改进。利用这一结果，Ｗｉｄｄｅｒ不等式得到了加强，并且建立了广义的Ｈａｒｄｙ－Ｌｉｔｔｌｅｗｏｏｄ不等式。     

Ou-Iang积分不等式的推广

本文推广了Ou-Iang在文[1]中给出的一类积分不等式。     

Ou-Iang积分不等式的推广

本文推广了Ｏｕ－Ｉａｎｇ在文［１］中给出的一类积分不等式     

两个积分不等式的推广

应用交换积分次序方法和微分中值定理等原理,对两个积分不等式进行拓展,使得这两个积分不等式的适用范围更广,证明过程更为简洁.     

一个特定型定积分不等式的若干推广

通过适当构造辅助函数和应用牛顿—莱布尼兹公式、施瓦兹积分不等式,将一个特定型定积分不等式进行了推广.证明了只要被积函数在积分区间内存在零点,该特定型定积分不等式均成立,进而给出实例说明了该不等式成立的正确性.     

一类积分不等式的推广

对一些基本的积分不等式进行了推广，给出了含有n个无关变元的更广泛的非线性积分不等式．利用所得的不等式讨论了某些非线性积分方程解的有界性．     

关于一类Hilbert积分不等式的推广

本文通过引入参数p,q,a,b和t,推广了Hilbert积分不等式的等价形式.     

Hardy-Hilbert积分不等式的一个新推广及其应用

通过引入一个形如x1 x(x∈[0, ∞))的幂指函数建立了带权的Hardy-Hilbert积分不等式的新推广.并证明了系数(2)(sinπp)是最佳值.作为应用,给出了Hardy-Littlewood积分不等式的一个推广.     

Consistent first-return Riemann sums for Lebesgue integrals

U. B. Darji and M. J. Evans [1] showed previously that it is possible to obtain the integral of a Lebesgue integrable function on the interval [0,1] via a Riemann type process, where one chooses the selected point in each partition interval using a first-return algorithm based on a sequence {x _n} which is dense in [0,1]. Here we show that if the same is true for every rearrangement of {x _n}, then the function must be equal almost everywhere to a Riemann integrable function. This revised version was published online in June 2006 with corrections to the Cover Date.     

Generalization of different type integral inequalities for s-convex functions via fractional integrals

In this paper, a general integral identity for twice differentiable functions is derived. By using of this identity, the author establishes some new estimates on Hermite-Hadamard type and Simpson type inequalities for s-convex via Riemann–Liouville fractional integral.     

Some inequalities of Qi type for double integrals

In the paper, the authors establish some new inequalities of Qi type for double integrals on a rectangle, from which some known integral inequalities of Qi type may be derived.     

New general extensions of Chebyshev type inequalities for Sugeno integrals

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

Riemann积分的历史与发展

总结Riemann积分的发展历史、基本思想及其各种推广,介绍Riemann积分的局限性和缺陷,以及由此推动产生Lebesgue积分的过程.     

涉及广义积分及其离散模型的某些非线性不等式

§ 1　IntroductionAs well-known thatthe theory of Gronwall-Bellman type integral inequalities,both inone and more than one variable,is crucial and indispensable in the development of almostall kinds of analytic equations such as differential and integral equations,integro-differen-tial equations,functional-differential equations and evolution equations.However,only afew investigation works appreared so far in the literature which dealt with Gronwall typeinequalities involving improper integra…     

On the limits of generalization of the Kolmogorov integral

The generalization of the Kolmogorov integral to functions with values in a Banach space is considered. It is proved that the resulting integral turns out to be essentially more general than the Bochner integral and is exactly equivalent to an integral of McShane type, whose definition requires that the scaling function be measurable.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 258–272.Original Russian Text Copyright © 2005 by A. P. Solodov.This revised version was published online in April 2005 with a corrected issue number.     

黎曼积分的完备化

综述了黎曼可积函数的基本特征,并指出黎曼可积函数列的极限运算在积分意义下是不封闭的.在构造了完备化空间之后,证明了该空间就是勒贝格可积函数空间,从而说明了黎曼积分的完备化形式是勒贝格积分.