Simpson and Newton type inequalities for convex functions via newly defined quantum integrals

We first establish two new identities, based on the kernel functions with either two section or three sections, involving quantum integrals by using new definition of quantum derivative. Then, some new inequalities related to Simpson's 1/3 formula for convex mappings are provided. In addition, Newton type inequalities, for functions whose quantum derivatives in modulus or their powers are convex, are deduced. We also mention that the results in this work generalize inequalities given in earlier study.     

Quantum Measurements and Kolmogorov's Theory of Probability

A connection between the probability space measurability requirement and the complementarity principle in quantum mechanics is established. It is shown that measurability of the probability space implies that the results of the quantum measurement depend not only on properties of the quantum object under consideration but also on classical characteristics of the measuring instruments used. It is also shown that if the measurability requirement is taken into account, then the hypothesis that the objective reality exists does not lead to the Bell inequality.     

On several new Hilbert-type inequalities involving means operators

In this paper, we establish several new Hilbert-type inequalities with a homogeneous kernel, involving arithmetic, geometric, and harmonic mean operators in both integral and discrete case. Such inequalities are derived by virtue of some recent results regarding general Hilbert-type inequalities and some well-known classical inequalities. We also prove that the constant factors appearing in established inequalities are the best possible. As an application, we consider some particular settings and compare our results with previously known from the literature.     

Functionals for multilinear fractional embedding

A novel representation is developed as a measure for multilinear fractional embedding.Corresponding extensions are given for the Bourgain–Brezis–Mironescu theorem and Pitt's inequality. New results are obtained for diagonal trace restriction on submanifolds as an application of the Hardy–Littlewood–Sobolev inequality. Smoothing estimates are used to provide new structural understanding for density functional theory, the Coulomb interaction energy and quantum mechanics of phase space. Intriguing connections are drawn that illustrate interplay among classical inequalities in Fourier analysis.     

A note on Hardy-type inequalities

We use a theorem of Cartlidge and the technique of Redheffer's ``recurrent inequalities" to give some results on inequalities related to Hardy's inequality.

     

Tomography of Spin States, the Entanglement Criterion, and Bell's Inequalities

We review the method of spin tomography of quantum states in which we use the standard probability distribution functions to describe spin projections on selected directions, which provides the same information about states as is obtained by the density matrix method. In this approach, we show that satisfaction or violation of Bell's inequalities can be understood as properties of tomographic functions for joint probability distributions for two spins. We compare results obtained using the methods of classical probability theory with those obtained in the framework of traditional quantum mechanics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 172–185, January, 2006.     

关于广义变分不等式与广义拟变分不等式

本文在非常一般的框架下,建立了极大极小不等式,广义变分不等式和广义拟变分不等式,证明了解的存在定理,且它们是在非紧集上得到的,从而推广和改进了[3～13]中的相应结果.     

短阵乘积和重排不等式的推广

In this paper, some inequalities for sequence rearrangement and matrix product areestablished. The authors extend and improve some known results, and show that there aresome errors in reference on inequalities for sequence rearrangement by examples,     

Some Hardy-Type Inequalities

In this paper, some new Hardy-type inequalities involving many functions are obtained. These on the one hand generalize and on the other hand improve some existing results by Isumi and Isumi, Levinson, and Pachpatte on this famous type of inequalities.     

离散型Carlson's不等式的一些推广及改进

In this paper, we consider Carlson type inequalities and discuss their possible improvement. First, we obtain two different types of generalizations of discrete Carlson's inequality by using the Hlder inequality and the method of real analysis, then we combine the obtained results with a summation formula of infinite series and some Mathieu type inequalities to establish some improvements of discrete Carlson's inequality and some Carlson type inequalities which are equivalent to the Mathieu type inequalities. Finally, we prove an integral inequality that enables us to deduce an improvement of the Nagy-Hardy-Carlson inequality.     

Multiplicative convexity and its applications

This paper is concerned with some sequences based on multiplicative convexity. Our results yield a class of new inequalities between ratios and differences of means, some of which extend the known ones.     

一类量子Markov半群的超压缩性与对数Sobolev不等式

本文在有限von Neumann代数生成的非交换概率空间L~p(p≥1)框架下,证明了一类量子Markov半群的超压缩性等价于其对应的Dirichlet型满足对数Sobolev不等式.此结果包含前人的相关成果为特例.作为推论,细化了Biane的相关工作.     

关于一个Hilbert类不等式及其应用

本文给出一个具有同样最佳常数π的Hilbert类不等式。作为应用，建立其等价形式及导出一些特殊的结果。     

关于n个无关变元的欧阳型离散型不等式

本文讨论关于ｎ个无关变元的Ｏｕ－Ｉａｎｇ型离散型不等式，所得结论是杨恩浩的Ｎ１上的Ｏｕ－Ｉａｎｇ型离散型不等式的自然推广和改进．     

一个较为精密的Hardy-Hilbert型不等式及其应用

利用改进的Euler-Maclaurin求和公式,求出一个较为精密的Hardy-Hilbert 型不等式．作为应用,给出它的等价形式及一些特殊结果．本文的引理及定理对研究这一类不等式提供了一些新的估算方法．     

加权平均值不等式推广到无穷多个数的形式

本文把有限个数的调和、几何、算术加权平均值不等式推广到无穷多个数的形式和广义积分的形式。     

Hardy-Hilbert不等式与Mulholland不等式的一个联系

本文引入单参数λ,利用β函数,建立Hardy-Hilbert不等式与Mulholland不等式的一个具有最佳常数因子的联系式．作为应用,给出它的等价形式及一些特殊结果．     

Short note on Hilbert’s inequality

Considering five different parameters, we obtain some new Hilbert-type integral inequalities for functions f(x), g(x) in L²[0, ∞). Then, we extract from our results some special cases which have been proved before.     

Beckner Inequality on Finite- and Infinite-Dimensional Manifolds

By using the dimension-free Harnack inequality, the coupling method, and Bakry-Emery's argument, some explicit lower bounds are presented for the constant of the Beckner type inequality on compact manifolds. As applications, the Beckner inequality and the transportation cost inequality are established for a class of continuous spin systems. In particular, some results in [1, 2] are generalized.     

关于Mill比的两个新结果

利用几何凸函数的性质,给出关于Mill比R x=ex22∫x+∞e-t22dt的两个新结果.当x>2.05时,它们都强于H.O.Pollak、A.V.Boyd得到的一些著名的结果.