Bo-Yan Xi, Fuzhen Zhang
The purpose of this paper is two-fold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua-Marcus' inequalities. Our results are stronger and more general than the existing ones.
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This note revisits some majorization inequalities for eigenvalues, special attention is given to an elegant theorem of Hiroshima. An extension of the special case of Hiroshima's theorem is presented. Some discussion and open problems are also included.
We study an elementary inequality supporting the classical Hermite-Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such new Schatten p-norm estimates and new majorization
In an Euclidean Jordan algebra V of rank n, an element x is said to be majorized by an element y, if the corresponding eigenvalue vector of x is majorized by the eigenvalue vector of y in R^n. In this article, we describe pointwise majorization inequalities of the form `T(x) majorized by S(x)', where T and S are linear transformations induced by Schur products. Specializing, we recover analogs of majorization inequalities of Schur, Hadamard, and Oppenheimer stated in the setting of Euclidean Jordan algebras, as well as majorization inequalities connecting quadratic and Lyapunov transformations…
Following the recent work of Jiang and Lin (Linear Algebra Appl. 585 (2020) 45--49), we present more results (bounds) on Harnack type inequalities for matrices in terms of majorization (i.e., in partial products) of eigenvalues and singular values. We discuss and compare the bounds derived through different ways. Jiang and Lin's results imply Tung's version of Harnack's inequality (Proc. Amer. Math. Soc. 15 (1964) 375--381); our results %with simpler proofs are stronger and more general than Jiang and Lin's. We also show some majorization inequalities concerning Cayley transforms. Some open pr…
We extend the notion of the determinant function $Λ$, originally introduced by T.Fack for $τ$-compact operators, to a natural algebra of $τ$-measurable operators affiliated with a semifinite von Neumann algebra which coincides with that defined by Haagerup and Schultz in the finite case and on which the determinant function is shown to be submultiplicative. Application is given to Hölder type inequalities via general Araki-Lieb-Thirring inequalities due to Kosaki and Han and to a Weyl-type theorem for uniform majorization.
Denote by $¶_n$ the set of $n\times n$ positive definite matrices. Let $D = D_1\oplus \dots \oplus D_k$, where $D_1\in ¶_{n_1}, \dots, D_k \in ¶_{n_k}$ with $n_1+\cdots + n_k=n$. Partition $C\in ¶_n$ according to $(n_1, \dots, n_k)$ so that $\Diag C = C_1\oplus \dots \oplus C_k$. We prove the following weak log majorization result: \begin{equation*} λ(C^{-1}_1D_1\oplus \cdots \oplus C^{-1}_kD_k)\prec_{w \,\log} λ(C^{-1}D), \end{equation*} where $λ(A)$ denotes the vector of eigenvalues of $A\in \Cnn$. The inequality does not hold if one replaces the vectors of eigenvalues by the vectors of sing…
We introduce a transport-majorization argument that establishes a majorization in the convex order between two densities, based on control of the gradient of a transportation map between them. As applications, we give elementary derivations of some delicate Fourier analytic inequalities, which in turn yield geometric "slicing-inequalities" in both continuous and discrete settings. As a further consequence of our investigation we prove that any strongly log-concave probability density majorizes the Gaussian density and thus the Gaussian density maximizes the Rényi and Tsallis entropies of all o…
We obtain some inequalities which are stronger than the Schur majorization inequalities.
Heron-Wasserstein majorization inequalities for spectral and Kubo-Ando geometric means
本文证明了谱几何平均和Kubo-Ando几何平均的Heron型优超不等式，并解决了Bhatia关于弱优超精炼的问题。
A log-majorization inequality for normal matrices with applications to determinantal inequalities and geometric means
研究建立正规矩阵对数优超不等式，扩展厄米特结果。通过该不等式解决行列式猜想正规矩阵扩展，给出加权几何均值乘积特征值图景，完善相关定理。
Several matrix/operator inequalies are given. Most of them are unexpected extensions of the Araki Log-majorization theorem, obtained thanks to a new log-majorization for positive linear maps and normal operators (Theorem 2.9). The main idea and technical tool is a two variables log-convex norm functional (Theorem 1.2).
Let $0<p<q\leq\infty$ and $α\in (0,\infty]$. We give a characterization of quasi-Banach interpolation spaces for the couple $(L_p(0,α),L_q(0,α))$ in terms of two monotonicity properties, extending known results which mainly dealt with Banach spaces. This enables us to recover recent results of Cwikel and Nilsson on sequence spaces and to solve a conjecture of Levitina, Sukochev and Zanin in the setting of function spaces. We apply the results obtained to characterize symmetric spaces in which the standard forms of the noncommutative Khintchine inequalities hold.
Motivated by Horn's log-majorization (singular value) inequality $s(AB)\underset{log}{\prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)\underset{w}{\prec} s(A)*s(B)$ for square complex matrices, we consider their Hermitian analogs $λ(\sqrt{A}B\sqrt{A}) \underset{log}{\prec} λ(A)*λ(B)$ for positive semidefinite matrices and $λ(|A\circ B|) \underset{w}{\prec} λ(|A|)*λ(|B|)$ for general (Hermitian) matrices, where $A\circ B$ denotes the Jordan product of $A$ and $B$ and $*$ denotes the componentwise product in $R^n$. In this paper, we extended these inequalities to the setting…
From Majorization to Horn and Unitary-Orbit Inequalities
本文提出一个统一框架，将优超不等式提升为完整的Horn不等式（等价于酉轨道不等式），并应用于三类算子不等式，得到推广的酉轨道不等式。
We prove majorization inequalities for different means of positive definite matrices. These include the Cartan mean (the Karcher mean), the log Euclidean mean, the Wasserstein mean and the power mean.
Weak log-majorization between the geometric and Wasserstein means
该文研究正定埃尔米特矩阵的多种均值间的对数优超关系，证明了正定矩阵乘积与其几何均值的奇异值对数优超，建立双变量Wasserstein均值与谱几何均值的弱对数优超，并验证特定条件下前者递减收敛于对数欧氏均值。
Log-majorization and matrix norm inequalities with application to quantum information
该研究推广了Araki对数优超，将其应用于量子信息中的α-z-Rényi散度，分析了多元Golden--Thompson型迹不等式与Karcher均值范数不等式的等号情形，并修正了此前加权几何均值范数不等式等号结果的证明。
We extend Ando-Hiai's log-majorization for the weighted geometric mean of positive definite matrices into that for the Cartan barycenter in the general setting of probability measures on the Riemannian manifold of positive definite matrices equipped with trace metric. The main key is the settlement of the monotonicity problem of the Cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone. We also derive a version of Lie-Trotter formula and related unitarily invariant norm inequalities for the Cartan barycenter as the ma…
Sub-additive and super-additive inequalities for concave and convex functions have been generalized to the case of matrices by several authors over a period of time. These lead to some interesting inequalities for matrices, which in some cases coincide with, and in other cases are at variance with the corresponding inequalities for real numbers. We survey some of these matrix inequalities and do further investigations into these. We introduce the novel notion of dominated majorization between the spectra of two Hermitian matrices $B$ and $C$, dominated by a third Hermitian matrix $A$. Based on…
For $α,z>0$ with $α\ne1$, motivated by comparison between different kinds of Rényi divergences in quantum information, we consider log-majorization between the matrix functions \begin{align*} P_α(A,B)&:=B^{1/2}(B^{-1/2}AB^{-1/2})^αB^{1/2}, \\ Q_{α,z}(A,B)&:=(B^{1-α\over2z}A^{α\over z}B^{1-α\over2z})^z \end{align*} of two positive (semi)definite matrices $A,B$. We precisely determine the parameter $α,z$ for which $P_α(A,B)\prec_{\log}Q_{α,z}(A,B)$ and $Q_{α,z}(A,B)\prec_{\log}P_α(A,B)$ holds, respectively.
The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by R. Lemos and G. Soares in \cite{lemos}. In addition, we present a complement of a unitarily invariant norm inequality which was conjectured by R. Bhatia, Y. Lim and T. Yamazaki in \cite{Bhatia2}, and recently proved by T.H. Dinh, R. Dumitru and J.A. Franco in \cite{Dinh} for the Schatten p-norm with $1\leq p\leq \infty$.
Fe{j}ér and Levin-Stečkin inequalities treat integrals of the product of convex functions with symmetric functions. The main goal of this article is to present possible matrix versions of these inequalities. In particular, majorization results are shown of Fejér type for both convex and log-convex functions. For matrix Levin-Stečkin type, we present more rigorous results involving the partial Loewner ordering for Hermitian matrices.
We offer a new proof of uniform convexity inequalities for the Finsler manifold of nonpositive curvature taken on the space of positive-semidefinite matrices with the weighted matrix geometric mean defining the geodesic between two points. Using the technique of log majorization, we are able to characterize that the equality cases of said equalities occur if and only if the matrices commute, and hence are the same as in $\ell^p$.
We consider positive, integral-preserving linear operators acting on $L^1$ space, known as stochastic operators or Markov operators. We show that, on finite-dimensional spaces, any stochastic operator can be approximated by a sequence of stochastic integral operators (such operators arise naturally when considering matrix majorization in $L^1$). We collect a number of results for vector-valued functions on $L^1$, simplifying some proofs found in the literature. In particular, matrix majorization and multivariate majorization are related in $\mathbb{R}^n$. In $\mathbb{R}$, these are also equiva…
We prove some sufficient conditions implying $l^p$ inequalities of the form $||x||_p \leq ||y||_p$ for vectors $ x, y \in [0,\infty)^n$ and for $p$ in certain positive real intervals. Our sufficient conditions are strictly weaker than the usual majorization relation. The conditions are expressed in terms of certain homogeneous symmetric polynomials in the entries of the vectors. These polynomials include the elementary symmetric polynomials as a special case. We also give a characterization of the majorization relation by means of symmetric polynomials.
Weak Log-Majorization for Negative Lim-Pálfia Power Means
本文研究负阶Lim-Pálfia幂均值的弱对数优超性，证明Ando的酉不变范数不等式对0<q<1的Schatten q-拟范数成立，解决了Hiai与Lim提出的问题，得到有限族正定矩阵的相关Schatten拟范数不等式。
Weak majorization inequalities for the cubic and quartic coefficients of $e^{(A+B)t}$ versus $e^{At}e^{Bt}$
本文研究了 $e^{(A+B)t}$ 与 $e^{At}e^{Bt}$ 的立方和四次系数的弱主化不等式，结合Ky Fan原理和交换子恒等式，证明了特征值弱主化关系并提出更高阶次的充分条件。
In this paper we show that the eigenvalue map and the symplectic eigenvalue map of positive definite matrices are Lipschitz for the Cartan-Hadamard Riemannian metric, and establish log-majorizations for the (symplectic) eigenvalues of the Cartan barycenter of integrable probability Borel measures. This leads a version of Jensen's inequality for geometric integrals of matrix-valued integrable random variables.
Majorization is a basic concept in matrix theory that has found applications in numerous settings over the past century. Power majorization is a more specialized notion that has been studied in the theory of inequalities. On the other hand, the trumping relation has recently been considered in quantum information, specifically in entanglement theory. We explore the connections between trumping and power majorization. We prove an analogue of Rado's theorem for power majorization and consider a number of examples.
Let $\mathcal{A}$ be a $C^*$-algebra and $ϕ:\cA\to L(H)$ be a positive unital map. Then, for a convex function $f:I\to \mathbb{R}$ defined on some open interval and a self-adjoint element $a\in \mathcal{A}$ whose spectrum lies in $I$, we obtain a Jensen's-type inequality $f(ϕ(a)) \leq ϕ(f(a))$ where $\le$ denotes an operator preorder (usual order, spectral preorder, majorization) and depends on the class of convex functions considered i.e., operator convex, monotone convex and arbitrary convex functions. Some extensions of Jensen's-type inequalities to the multi-variable case are considered.
We prove a moment majorization principle for matrix-valued functions with domain $\{-1,1\}^{m}$, $m\in\mathbb{N}$. The principle is an inequality between higher-order moments of a non-commutative multilinear polynomial with different random matrix ensemble inputs, where each variable has small influence and the variables are instantiated independently. This technical result can be interpreted as a noncommutative generalization of one of the two inequalities of the seminal invariance principle of Mossel, O'Donnell and Oleszkiewicz. Applications to noncommutative noise stability and noncommutati…
We present several matrix and operator inequalities of Hermite-Hadamard type. We first establish a majorization version for monotone convex functions on matrices. We then utilize the Mond-Pecaric method to get an operator version for convex functions. We also present some applications. Finally we obtain an Hermite-Hadamard inequality for operator convex functions, positive linear maps and operators acting on Hilbert spaces.
Given two positive definite matrices $A$ and $B$, a well known result by Gelfand, Naimark and Lidskii establishes a relationship between the eigenvalues of $A$ and $B$ and those of $AB$ by means of majorization inequalities. In this work we make a local study focused in the spectrum of the matrices that achieve the equality in those inequalities. As an application, we complete some previous results concerning Procustes problems for unitarily invariant norms in the manifold of positive definite matrices.
Majorization in some symplectic weak supermajorizations
本文研究辛特征值的弱超优化关系，建立了通过优化使其达到饱和的充分必要条件。
Sharp Hyperbolic Cutoffs and Dimension-Sharp Counterexamples for Reverse Araki-Type Inequalities
研究指数超过2的反向Araki型迹不等式和对数优超，发现s=2是尖锐无维边界，并给出3×3正定矩阵反例，同时刻画了幂权重的尖锐双曲边界。
Aggregate Bounds on the eigenvalues of the principal submatrices of a Hermitian matrix and majorization relations
本文扩展了Thompson关于Hermitian矩阵主子矩阵特征值和的界，并利用该结果推导出所有主子矩阵特征值之间的主导关系，涵盖Schur和Szasz的已有结果。
Log-majorizations between quasi-geometric type means for matrices
本文研究了矩阵准几何型均值之间的对数支配关系，并探讨了相关迹函数的联合凹凸性。
Some inequalities for spectral geometric mean with applications
本文研究谱几何平均的不等式，推导了正可逆算子的Hölder型不等式、算子序关系及范数版本，证明矩阵情形的对数优超关系，得到量子Tsallis相对熵的序关系与新下界。
In information theory, the well-known log-sum inequality is a fundamental tool which indicates the non-negativity for the relative entropy. In this article, we establish a set of inequalities which are similar to the log-sum inequality involving two functions defined on scalars. The parametric extended log-sum inequalities are shown. We extend these inequalities for the commutative matrices. In addition, utilizing the Löwner partial order relation and the Hansen-Pedersen theory for non-commutative positive semi-definite matrices we demonstrate a number of matrix-inequalities analogous to the l…
In this paper, we introduce the notion of log-φ-convex functions and present some properties and representation of such functions. We obtain some results of the Hermite Hadamard inequalities for product log-φ-convex functions.
The aim of this paper is to study the Harnack type logarithmic submajorisation and Fuglede-Kadison determinant inequalities for operators in a finite von Neumann algebra. In particular, the Harnack type determinant inequalities due to Lin-Zhang [15] and Yang-Zhang [27] are extended to the case of operators in a finite von Neumann algebra.
Mixed $f$-divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log concave functions and establish some of their properties. Among them are affine invariant vector entropy inequalities, like new Alexandrov-Fenchel type inequalities and an affine isoperimetric inequality for the vector form of the Kullback Leibler divergence for log concave functions. Special cases of $f$-divergences are mixed $L_λ$-affine surface areas for log concave functions. For those, we establish various affine isoperimetric i…
Motivated by some recently established operator Jensen-type inequalities related to a usual convexity, in the present paper we derive several more accurate operator Jensen-type inequalities for certain subclasses of convex functions. More precisely, we obtain interpolating series of Jensen-type inequalities for log-convex and non-negative superquadratic functions. In particular, we obtain the corresponding refinements of the Jensen-Mercer operator inequality for such classes of functions.
In this paper, we extend some estimates of the left hand side of a Hermite- Hadamard type inequality for nonconvex functions whose derivatives absolute values are preinvex and log-preinvex.
Improvements on some partial trace inequalities for positive semidefinite block matrices
本文研究半正定分块矩阵的偏迹不等式，用新方法证明并推广 Choi 的结果，改进 Li-Liu-Huang 的结论，并给出优化、酉不变范数和奇异值相关不等式。
Equality in some symplectic eigenvalue inequalities
本文研究辛特征值不等式的取等情形，给出辛形式 Weyl、Lidskii 及 Schur–Horn 控制不等式等号成立的充要条件，并证明前两者与经典特征值等号条件相似。
Isospectral majorization and isoperimetric inequalities for coherent states on the Bloch sphere
该研究针对布洛赫球上的SU(2)相干态，证明Lieb-Solovej不等式的等谱版本，应用得到Toeplitz算子相关等周不等式，未用球面等周不等式。
In this paper, we establish some new Hadamard type inequalities using elementary well known inequalities for functions whose inequalities absolute values are α-, m-, (α,m)-logarithmically convex.
Stability under mixtures of transportation inequalities and restricted log-Sobolev inequalities
本文研究传输-熵不等式与受限泛函不等式在混合下的稳定性，给出将不等式从混合分量转移到混合本身的充分条件。
We review some simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality.
This paper has been withdrawn by the author
Eigenvalues inequalities involving (log) convex/concav functions and Hermitian matrices, positive unital maps are considered. Simple proofs of Bhatia-Kittaneh inequality and Naimark dilation theorem are given.
Some inequalities for log-convex functions of selfadjoint operators on quaternionic Hilbert spaces
本文针对四元数希尔伯特空间上的自伴算子，利用对数凸函数证明了若干Jensen型不等式，并通过特定对数凸函数得到算子不等式的特殊情况，丰富了该领域的不等式研究。
We prove logarithmic Sobolev inequalities for semi-direct product operators (see definition in Section 1). We apply our main results to examples of operators and provide some applications to ultracontractive bounds of semigroups. Hardy's inequalities are proved and use to study some operators of Grushin's type.
In this paper, we describe s-logarithmically convex functions in the first and second sense which are connected with the ordinary logatihmic convex and s-convex in the first and second sense. Afterwards, some new inequalities related to above new definitions are given.
We present new proofs of two theorems of E.B. Davies and B. Simon about ultracontractivity property for of semigroups of operators and logarithmic Sobolev inequalities with parameter (LSIWP for short) satisfied by the generator of the semigroup. In our proof, we use neither the $L^p$ version of the $LSIWP$ nor Stein's interpolation. Our tool is Nash type inequality. We also present new results. In particular, we show that Nash type inequality and LSIWP are equivalent.
We propose a new, self-contained, approach to H. Raufi's extension of Prekopa's theorem for matrix-valued log-concave functions. Along the way, new related inequalities are established, in particular a Brascamp-Lieb variance inequality for matrix weights.
We prove, using optimal transport tools, weighted Poincar'e inequalities for log-concave random vectors satisfying some centering conditions. We recover by this way similar results by Klartag and Barthe-Cordero-Erausquin for log-concave random vectors with symmetries. In addition, we prove that the variance conjecture is true for increments of log-concave martingales.
In this paper, we establish some new Hadamard type inequalities for s-logarithmically convex functions in the second sense via fractional integrals by using Lemma 1 which has been proved by Sarikaya et al. in the paper [3].
We study a class of logarithmic Sobolev inequalities with a general form of the energy functional. The class generalizes various examples of modified logarithmic Sobolev inequalities considered previously in the literature. Refining a method of Aida and Stroock for the classical logarithmic Sobolev inequality, we prove that if a measure on $\mathbb{R}^n$ satisfies a modified logarithmic Sobolev inequality then it satisfies a family of $L^p$-Sobolev-type inequalities with non-Euclidean norms of gradients (and dimension-independent constants). The latter are shown to yield various concentration-…
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