On sharp triangle inequalities in Banach spaces

We shall present a couple of norm inequalities which will much improve the sharp triangle inequality with n elements and its reverse inequality in a Banach space shown recently by the last three authors.     

Reverses of the continuous triangle inequality for Bochner integral in complex Hilbert spaces

Some reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in complex Hilbert spaces are given. Applications for complex-valued functions are provided as well.     

Some inequalities in inner product spaces related to the generalized triangle inequality

In this paper we obtain some inequalities related to the generalized triangle and quadratic triangle inequalities for vectors in inner product spaces. Some results that employ the Ostrowski discrete inequality for vectors in normed linear spaces are also obtained.     

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.     

Higher order Ostrowski type inequalities over Euclidean domains

The classical Ostrowski inequality for functions on intervals estimates the value of the function minus its average in terms of the maximum of its first derivative. This result is extended to functions on general domains using the L^∞ norm of its nth partial derivatives. For radial functions on balls the inequality is sharp.     

On Pólya–Szegö reverse of Schwarz's inequality

We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner product spaces. Applications to the famous Hadamard's inequality about determinants and the triangle inequality for norms are given.     

Inequalities on the Triangle

The article deals with generalizations of the inequalities for convex functions on the triangle. The Jensen and the Hermite-Hadamard inequality are included in the study. Considering a convex function on the triangle, we obtain a generalization of the Jensen-Mercer inequality, and a refinement of the Hermite-Hadamard inequality.     

An elementary proof of the triangle inequality for the Wasserstein metric

We give an elementary proof for the triangle inequality of the -Wasserstein metric for probability measures on separable metric spaces. Unlike known approaches, our proof does not rely on the disintegration theorem in its full generality; therefore the additional assumption that the underlying space is Radon can be omitted. We also supply a proof, not depending on disintegration, that the Wasserstein metric is complete on Polish spaces.

     

子空间非相似性度量(WWF-SSD)的三角不等式

2005年,W ang,W ang和Feng在国际期刊Pattern R ecogn ition上提出了用来刻画子空间非相似性的度量(WW F-SSD),而且利用这个度量设计人脸识别算法,取得了很好的效果.但是该文章中没能证明WW F-SSD满足一个距离必须具有的三角不等式性质.本文给出了WW F-SSD的一个矩阵形式等价定义.基于这个定义不但可以使利用M atL ab实现的算法更有效率,而且可以很直观的证明WW F-SSD不依赖于子空间标准正交基的选择这一性质.进一步,我们在这个定义的基础上,利用矩阵的有关性质证明了WW F-SSD的三角不等式,从而最终证明了WW F-SSD是距离.     

On new generalizations of the Hilbert integral inequality

Some new generalizations of the Hilbert integral inequality by introducing real functions ?(x) and ψ(x). The results of this paper reduce to those of the corresponding inequalities proved by Gao [Mingzhe Gao, On Hilbert's integral inequality, Math. Appl. 11 (3) (1998) 32-35]. Some applications are considered.     

Vanishing theorems for ach Kähler manifolds and harmonic maps

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L ² cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincaré inequality.     

Hypercontractivity for log-subharmonic functions

We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on Rn and different classes of measures: Gaussian measures on Rn, symmetric Bernoulli and symmetric uniform probability measures on R, as well as their convolutions. Surprisingly, a slightly weaker strong hypercontractivity property holds for any symmetric measure on R. A log-Sobolev inequality (LSI) is deduced from the (SHC) for compactly supported measures on Rn, still for log-subharmonic functions. An analogous (LSI) is proved for Gaussian measures on Rn and for other measures for which we know the (SHC) holds. Our log-Sobolev inequality holds in the log-subharmonic category with a constant smaller than the one for Gaussian measure in the classical context.     

The logarithmic HLS inequality for systems on compact manifolds

We prove an optimal logarithmic Hardy-Littlewood-Sobolev inequality for systems on compact m-dimensional Riemannian manifolds, for any m?2. We show that a special case of the inequality, involving only two functions, implies the general case by using an argument from the theory of linear programing.     

Reinforcement of an inequality due to Brascamp and Lieb

In this paper, we obtain a reinforcement of an inequality due to Brascamp and Lieb and a reinforcement of Poincaré's inequality for general logarithmical concave measures on Rd. The formula used in the proof is related to theorems concerning the integration of log-concave functions (such as results of Prékopa and of Ball, Barthe and Naor). We also obtain a lower bound for the variance of the same family of measures.     

Cyclically antimonotone vector equilibrium problems

ABSTRACT

In this paper, we extend the notion of cyclic antimonotonicity (known for scalar bifunctions) to the vector case, in order to obtain a vectorial equilibrium version of Ekeland's variational principle. We characterize the cyclic antimonotonicity in terms of a suitable approximation from below of the vector bifunction, which allows us to avoid the demanding triangle inequality property, usually required in the literature, when dealing with Ekeland's principle for bifunctions. Furthermore, a result for weak vector equilibria in the absence of convexity assumptions is given, without passing through the existence of approximate solutions.     

Notes on the Norm Estimates for the Sum of Two Matrices

This is a lecture note of my joint work with Chi-Kwong Li concerning various results on the norm structure of n x n matrices (as Hilbert-space operators). The main result says that the triangle inequality serves as the ultimate norm estimate for the upper bounds of summation of two matrices.In the case of summation of two normal matrices, the result turns out to be a norm estimate in terms of the spectral variation for normal matrices.     

Log-convexity and log-concavity of hypergeometric-like functions

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms a?∑fkk(a)xk, a?∑fkΓ(a+k)xk and a?∑fkxk/k(a). The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Turán inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexity results for the same function recently proved by Baricz. Besides, we establish a reverse inequality which complements naturally the inequality of Barnard, Gordy and Richards. Similar results are established for the Gauss and the generalized hypergeometric functions. A conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made.     

Refinement of Hardy's inequalities via superquadratic and subquadratic functions

A new refined weighted Hardy inequality for p?2 is proved and discussed. The inequality is reversed for 1<p?2, which means that for p=2 we have equality. The main tool in the proofs are some new results for superquadratic and subquadratic functions.     

两个新的三角不等式及应用

给出两个新的三角不等式 ,并将其应用于讨论角成等比的三角形形状 .     

Duality in ε-variational inequality problems

In this paper a general analysis of duality for an extended ε-variational inequality problem based on the notions of ε-convexity and ε-conjugacy is performed. Optimal solutions of both the primal and dual problems are also related to the saddle point of an associated Lagrangian. Gap functions for these problems are proposed. An existence theorem for the extended ε-variational inequality is also established by means of the KKM lemma.