Enhancements to the von Neumann trace inequality

Upper trace bounds for the product of two n × n complex matrices are presented. The real component of the trace inequality is tighter than von Neumann’s inequality, and the imaginary component is new.     

A Cauchy-Khinchin integral inequality

This paper discusses some Cauchy-Khinchin integral inequalities. Khinchin [2] obtained an inequality relating the row and column sums of 0-1 matrices in the course of his work on number theory. As pointed out by van Dam [6], Khinchin’s inequality can be viewed as a generalization of the classical Cauchy inequality. Van Dam went on to derive analogs of Khinchin’s inequality for arbitrary matrices. We carry this work forward, first by proving even more than general matrix results, and then by formulating them in a way that allows us to apply limiting arguments to create new integral inequalities for functions of two variables. These integral inequalities can be interpreted as giving information about conditional expectations.     

A matrix trace inequality and its application

In this short paper, we give a complete and affirmative answer to a conjecture on matrix trace inequalities for the sum of positive semidefinite matrices. We also apply the obtained inequality to derive a kind of generalized Golden-Thompson inequality for positive semidefinite matrices.     

Eigenvalue extensions of Bohr’s inequality

We present a weak majorization inequality and apply it to prove eigenvalue and unitarily invariant norm extensions of a version of the Bohr’s inequality due to Vasi? and Ke?ki?.     

A relationship between subpermanents and the arithmetic-geometric mean inequality

Using the arithmetic-geometric mean inequality, we give bounds for k-subpermanents of nonnegative n×n matrices F. In the case k=n, we exhibit an n²-set S whose arithmetic and geometric means constitute upper and lower bounds for per(F)/n!. We offer sharpened versions of these bounds when F has zero-valued entries.     

DDVV-type inequality for skew-symmetric matrices and Simons-type inequality for Riemannian submersions

In this paper, we will first derive a DDVV-type optimal inequality for real skew-symmetric matrices, then we apply it to establish a Simons-type integral inequality for Riemannian submersions with totally geodesic fibres and Yang–Mills horizontal distributions. In this way, we show phenomenons of duality between submanifold geometry and Riemannian submersion, particularly between second fundamental form of a submanifold and integrability tensor of a Riemannian submersion.     

An asymmetric Kadison’s inequality

Some inequalities for positive linear maps on matrix algebras are given, especially asymmetric extensions of Kadison’s inequality and several operator versions of Chebyshev’s inequality. We also discuss well-known results around the matrix geometric mean and connect it with complex interpolation.     

Generalized matrix version of reverse Hölder inequality

A generalized matrix version of reverse Cauchy-Schwarz/Hölder inequality is proved. This includes the recent results proved by Bourin, Fujii, Lee, Niezgoda and Seo.     

Extensions of Heinz-Kato-Furuta inequality

We give an extension of Lin's recent improvement of a generalized Schwarz inequality, which is based on the Heinz-Kato-Furuta inequality. As a consequence, we can sharpen the Heinz-Kato-Furuta inequality.

     

Operator extensions of Hua’s inequality

We give an extension of Hua’s inequality in pre-Hilbert C^∗-modules without using convexity or the classical Hua’s inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C^∗-modules, another extension of Hua’s inequality is obtained. We also present an operator Hua’s inequality, which is equivalent to operator convexity of given continuous real function.     

反向Dresher不等式

配合使用Houlder不等式和Minkowski不等式,得到了反向Dresher不等式。     

Lp-Brunn-Minkowski inequality

We introduce the notion of L_p-mixed intersection body (p < 1) and extend the classical notion dual mixed volume to an L_p setting. Further, we establish the Brunn-Minkowski inequality for the q-dual mixed volumes of star duals of L_p-mixed intersection bodies.     

A proof of Hessian Sobolev inequality

In this paper, taking the Hessian Sobolev inequality (0<p≤k) (X.-J. Wang, 1994 [2]) as the starting point, we give a proof of the Hessian Sobolev inequality when k<p≤k^∗, where k^∗ is the critical Sobolev embedding index of k-Hessian type. We also prove that k^∗ is optimal by one-dimensional Hardy’s inequality.     

An inequality for means with applications

We show that an almost trivial inequality between the first and second moment and the maximal value of a random variable can be used to slightly improve deep theorems. Received: 25 November 2006     

Two new forms of half-discrete Hilbert inequality

In this paper, we introduce two new forms of the half-discrete Hilbert inequality. The first form is a sharper form of the half-discrete Hilbert inequality and is related to Hardy inequality. In the second one, we give a differential form of this inequality.     

Operator Aczél inequality

We establish several operator versions of the classical Aczél inequality. One of operator versions deals with the weighted operator geometric mean and another is related to the positive sesquilinear forms. Some applications including the unital positive linear maps on C^*-algebras and the unitarily invariant norms on matrices are presented.     

The dual difference Aleksandrov–Fenchel inequality

Recently, a dual Minkowski inequality and a dual Brunn–Minkowski inequality for volume differences were established. Following this, in this paper we establish a dual Aleksandrov–Fenchel inequality for dual mixed volume differences which generalizes several recent results.     

On the converse Jensen inequality

We give a survey on the converse Jensen inequality and we show that several recently published inequalities are simple consequences of certain long time known results. We also give a new refinement of the converse Jensen inequality as well as improvements of some related results.     

Existence of extremal functions for a Hardy-Sobolev inequality

We present the best constant and the extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in RN.     

An order preserving inequality for three operators via furuta inequality

Furuta showed that if A≥B≥0,then for each r≥0,f（p）=（A^r/2 B^p A^r/2）^t＋r/p＋r is decreasing for p≥t≥0.Using this result,the following inequality（C^r/2（AB^2A）^δC^ r/2）^ p-1＋r/4δ＋r ≤C^p-1＋r is obtained for 0〈p ≤1,r≥1,1/4≤δ≤1 and three positive operators A, B, C satisfy（A^1/2BA^1/2）^p/2≤A^p,（B^1/2AB^1/2）^p/2≥B^p,（C^1/2AC^1/2）^p/2≤C^p,（A^1/2CA^1/2）^p/2≥A^p.