An extension of inequalities by Ando

Variants of Ando--Hiai type inequalities for deformed means and applications

For an $n$-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando--Hiai type inequalities for deformed means from an $n$-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

On Ando's inequalities for convex and concave functions

For positive semidefinite matrices $A$ and $B$, Ando and Zhan proved the inequalities $||| f(A)+f(B) ||| \ge ||| f(A+B) |||$ and $||| g(A)+g(B) ||| \le ||| g(A+B) |||$, for any unitarily invariant norm, and for any non-negative operator monotone $f$ on $[0,\infty)$ with inverse function $g$. These inequalities have very recently been generalised to non-negative concave functions $f$ and non-negative convex functions $g$, by Bourin and Uchiyama, and Kosem, respectively. In this paper we consider the related question whether the inequalities $||| f(A)-f(B) ||| \le ||| f(|A-B|) |||$, and $||| g(A…

Ando dilations and inequalities on noncommutative varieties

We obtain dilation results which simultaneously generalize Sz.-Nagy dilation theorem for contractions, Ando's dilation theorem for commuting contractions, Sz.-Nagy--Foias commutant lifting theorem, and Schur's representation for the unit ball of H^\infty, in the framework of noncommutative varieties in several variables and Poisson kernels on Fock spaces. This leads to inequalities which, in the particular case of two contractions, are sharper than Ando's inequality and Agler-McCarthy's inequality in the setting of commuting contractive matrices or arbitrary commuting contractions of class C_0…

Ando dilations and inequalities on noncommutative domains

We obtain intertwining dilation theorems for noncommutative regular domains D_f and noncommutative varieties V_J of n-tuples of operators, which generalize Sarason and Sz.-Nagy--Foias commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain D_f (resp. variety V_J) as well as a Schur type representation for the unit ball of the Hardy algebra associated with the variety V_J. We provide Ando type dilations and inequalities for bi-domains D_f \times D_g and bi-varieties V_J \times V…

谱几何平均与Kubo-Ando几何平均的Heron-Wasserstein优超不等式

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。

算子谱几何平均与几何平均的比较

Operator Spectral Geometric Versus Geometric Mean

本文研究希尔伯特空间上正定算子的谱几何平均，推导其新不等式，给出谱几何平均与常规几何平均的显式比较，并将结果应用于Ando型及Ando-Hiai不等式，补充了算子几何平均的相关不等式理论。

正规算子的根与安藤定理的推广

On roots of normal operators and extensions of Ando's Theorem

本文推广安藤定理至$k$-仿正规、绝对$k$-仿正规算子，且在可分希尔伯特空间中证明$k$-拟仿正规算子若其某幂次正规，则可分解为正规算子与幂零指数至多为$\min\{n,k+1\}$的幂零算子的直和。

Extensions of some matrix inequalities related to trace and partial traces

We first present a determinant inequality related to partial traces for positive semidefinite block matrices. Our result extends a result of Lin [Czech. Math. J. 66 (2016)] and improves a result of Kuai [Linear Multilinear Algebra 66 (2018)]. Moreover, we provide a unified treatment of a result of Ando [ILAS Conference (2014)] and a recent result of Li, Liu and Huang [Operators and Matrices 15 (2021)]. Furthermore, we also extend some determinant inequalities involving partial traces to a larger class of matrices whose numerical ranges are contained in a sector. In addition, some extensions on…

负Lim-Pálfia幂均值的弱对数优超性

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。

On norm sub-additivity and super-additivity inequalities for concave and convex functions

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…

Extensions of the Schur majorisation inequalities

We obtain some inequalities which are stronger than the Schur majorization inequalities.

Sobolev扩展、插值不等式及其推论

Sobolev extensions, interpolation inequalities and consequences

本文引入Lebesgue W^{1,p}扩展域概念，证明任何W^{1,p}扩展域（1<p<∞）都是Lebesgue W^{1,p}扩展域，并推导出类似全空间形式的Sobolev插值不等式，讨论其应用。

Some extensions of the Young and Heinz inequalities for Matrices

In this paper, we present some extensions of the Young and Heinz inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with matrices. More precisely, for two positive semidefinite matrices $A$ and $B$ we show that \begin{align*} \Big\|A^νXB^{1-ν}+A^{1-ν}XB^ν\Big\|_{2}^{2}\leq\Big\|AX+XB\Big\|_{2}^{2}- 2r\Big\|AX-XB\Big\|_{2}^{2}-r_{0}\left(\Big\|A^{\frac{1}{2}}XB^{\frac{1}{2}}-AX\Big\|_{2}^{2}+ \Big\|A^{\frac{1}{2}}XB^{\frac{1}{2}}-XB\Big\|_{2}^{2}\right), \end{align*} where $X$ is an arbitrary $n\times n$ matrix, $0<ν…

数值半径不等式的 Orlicz 推广

Orlicz extension of Numerical radius inequalities

该研究借助 Orlicz 函数与 Hermite-Hadamard 不等式，建立 Hilbert 空间算子的改进数值半径上界，并给出特定分块矩阵的数值半径上界。

Brunn-Minkowski and Zhang inequalities for Convolution Bodies

A quantitative version of Minkowski sum, extending the definition of $θ$-convolution of convex bodies, is studied to obtain extensions of the Brunn-Minkowski and Zhang inequalities, as well as, other interesting properties on Convex Geometry involving convolution bodies or polar projection bodies. The extension of this new version to more than two sets is also given.

Extension of Lipschitz-type operators on Banach function spaces

We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and other measure-theoretic notions are introduced.%more appropriate. Thus, we adapt the classical definition of Lipschitz map to the context of spaces of integrable functions by introducing such elements. We analyze Lipschitz type inequalities in two fundamental cases. The first concerns a.e. pointwise inequalities, while the second considers dominations involving …

An extension problem and Hardy's inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type

In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator. Next, we study the mapping properties of the extension operator. In the last part we prove Poincaré-Sobolev inequalities on these spaces.

Operator inequalities associated with the Kantrovich type inequalities for $s$-convex functions

In this paper, we prove some operator inequalities associated with an extension of the Kantorovich type inequality for $s$-convex function. We also give an application to the order preserving power inequality of three variables and find a better lower bound for the numerical radius of a positive operator under some conditions.

Some extensions of Berezin number inequalities on operators

In this paper, we establish some upper bounds for Berezin number inequalities including of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=\left[\begin{array}{cc} 0&X, Y&0 \end{array}\right]$, then \begin{align*} \textbf{ber}^{r}(T)\leq 2^{r-2}\left(\textbf{ber}(f^{2r}(|X|)+g^{2r}(|Y^*|))+\textbf{ber}(f^{2r}(|Y|)+g^{2r}(|X^*|))\right)\\ -2^{r-2} \inf_{\|(k_{λ_{1}},k_{λ_{2}})\|=1} η(k_{λ_{1}},k_{λ_{2}}), \end{align*} where $η(k_{λ_{1}}, k_{λ_{2}}) = \left(\left\langle(f^{2r}(|X|)+g^{2r}(|Y^*|)\right)k_{λ_{2}},k_{λ_{2}}\right\rangle^\f…

Extensions of the Inequalities of Hardy and Hilbert

In this note we produce generalized versions of the classical inequalities of Hardy and of Hilbert and we establish their equivalence. Our methods rely on the H^1-BMOA duality. We produce a class of examples to establish that the generalizations are non-trivial.

关于加权Blaschke–Santalo不等式与强Brascamp–Lieb不等式

On weighted Blaschke--Santalo and strong Brascamp--Lieb inequalities

本文研究函数型Blaschke-Santaló不等式的新推广，探索其在标准高斯测度经典设定外的应用，涉及加权Blaschke–Santalo不等式与强Brascamp–Lieb不等式。

p-进Rademacher混沌的Sidon型不等式

Sidon-type inequalities for $p$-adic analogues of Rademacher chaos

研究p-进Rademacher混沌的最一般扩展，证明了2d/(d+1)-Sidon不等式，揭示了该系统的结构特性。

Some extensions of the operator entropy type inequalities

In this paper, we establish some reverses of the operator entropy inequalities under certain conditions by using the Mond-Pečarić method. In particular, we present {\tiny \begin{align*} f&\left[\int_T(A_s\natural_{p+1}B_s)dμ(s)+t_0\left(I_{\mathscr H}-\int_TA_s\natural_pB_sdμ(s)\right)\right]-γ_ff(t_0)\left(I_{\mathscr H}-\int_TA_s\natural_pB_sdμ(s)\right)\nonumber\\ &\le γ_f\widetilde{S}_p^f(\mathbf{A}|\mathbf{B})\,, \end{align*}} where $T$ is a locally compact Hausdorff space and $μ$ is a Radon measure on $T$, $0<m A_s \leq B_s \leq M A_s\,\,(s\in T)$ for some positive real numbers $m, M$ su…

Extensions of the vector-valued Hausdorff-Young inequalities

In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff--Young, Hardy--Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff--Young and Hardy--Littlewood inequalities and state that the Fourier transform is bounded from $L^p(\mathbb{R}^d,|\cdot|^{βp})$ into $L^q(\mathbb{R}^d,|\cdot|^{-γq})$ under certain condition on $p,q,β$ and $γ$. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and rela…

Some extensions of the Prékopa-Leindler inequality using Borell's stochastic approach

We present an abstract form of the Prékopa-Leindler inequality that includes several known -and a few new- related functional inequalities on Euclidean spaces. The method of proof and also the formulation of the new inequalities are based on Christer Borell's stochastic approach to Brunn-Minkowski type inequalities.

一致有界余链扩展与一致庞加莱不等式

Uniformly Bounded Cochain Extensions and Uniform Poincaré Inequalities

本文提出一种新的全局有界余链扩展算子，用于Lipschitz域上的微分形式，恢复了外导数的全局可交换性，为CutFEM的严谨基础提供了缺失的分析工具，并获得了非凸域的连续一致庞加莱不等式和第一Neumann特征值下界。

调和延拓的精确量化积分不等式

Sharp quantitative integral inequalities for harmonic extensions

本文针对Poisson算子及其伴随算子，证明了Hang、Wang和Yan的精确积分不等式的量化版本，获得了最强范数和不一定等于2的最优稳定性指数。

Weighted variation inequalities for differential operators and singular integrals in higher dimensions

We prove weighted $q$-variation inequalities with $2<q<\infty$ for differential and singular integral operators in higher dimensions. The vector-valued extensions of these inequalities are also given.

时间尺度上反向Callebaut、Rogers Hölder和Cauchy Schwarz不等式的核型分数扩展

Kernel-Type Fractional Extensions of Reverse Callebaut, Rogers Hölder and Cauchy Schwarz Inequalities on Time Scales

研究时间尺度上反向Callebaut等不等式的核型分数扩展，借助与钻石-α积分相关的非负核算子，能统一处理连续、离散、量子和分数动态环境下的反向不等式，还可用于相关方程的先验估计。

变指数Lebesgue空间中的Marcinkiewicz-Zygmund不等式

Marcinkiewicz-Zygmund inequalities in variable Lebesgue spaces

该研究针对变指数Lebesgue空间上线性算子的ℓ^r值扩张问题，刻画了对应有界扩张存在的r的范围，分析了不同测度下的差异，并给出了相关应用。

Hermite-Hadamard-type inequalities in the approximate integration

We give a slight extension of the Hermite-Hadamard inequality on simplices and we use it to establish error bounds of the operators connected with the approximate integration.

On matrix-valued log-concavity and related Prekopa and Brascamp-Lieb inequalities

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.

Hausdorff-Young inequalities for nonunimodular groups

The paper studies Hausdorff-Young inequalities for certain group extensions, by use of Mackey's theory. We consider the case in which the dual action of the quotient group is free almost everywhere. This result applies to yield a Hausdorff-Young inequality for nonunimodular groups.

Caffarelli-Kohn-Nirenberg 与加权高斯 Poincaré 不等式：尖锐 $L^2$ 稳定性与 $L^p$ 扩展的完整刻画

Caffarelli-Kohn-Nirenberg and Weighted Gaussian Poincaré Inequalities: a complete characterization of sharp $L^2$ stability and $L^p$ extensions

引入新加权高斯Poincaré不等式族，通过广义拉盖尔多项式、球谐分解和开尔文变换，完全刻画了CKN不等式的L^2稳定性并扩展至L^p情形。

An operator extension of the parallelogram law and related norm inequalities

We establish a general operator parallelogram law concerning a characterization of inner product spaces, get an operator extension of Bohr's inequality and present several norm inequalities. More precisely, let ${\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a Radon measure $μ$ and let $(A_t)_{t\in T}$ be a continuous field of operators in ${\mathfrak A}$ such that the function $t \mapsto A_t$ is norm continuous on $T$ and the function $t \mapsto \|A_t\|$ is integrable. If $α: T \times T \to \mathbb{C}$ is a measurable function such that $\bar{α(t,s)}…

α-concave functions and a functional extension of mixed volumes

Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation + on the class of quasi-concave functions, such that every class of α-concave functions is closed under +. We then define the mixed integrals, which are the polarization of the integral with respect to +. We proceed to discuss the extension of various classic inequalities to the functional setting. For general quasi-conca…

When does $e^{-\lvert τ\rvert }$ maximize Fourier extension for a conic section?

In the past decade, much effort has gone into understanding maximizers for Fourier restriction and extension inequalities. Nearly all of the cases in which maximizers for inequalities involving the restriction or extension operator have been successfully identified can be seen as partial answers to the question in the title. In this survey, we focus on recent developments in sharp restriction theory relevant to this question. We present results in the algebraic case for spherical and hyperbolic extension inequalities. We also discuss the use of the Penrose transform leading to some negative an…

On Hadamard Type Inequalities Involving Several Kind of Convexity

In this paper, we not only give the extensions of the results given in [7] by Gill et al. for log-convex functions, but also obtain some new Hadamard type inequalities for log-convex, m-convex and (alpha,m)-convex functions.

Extension of Euclidean operator radius inequalities

To extend the Euclidean operator radius, we define $w_p$ for an $n$-tuples of operators $(T_1,\ldots, T_n)$ in $\mathbb{B}(\mathscr{H})$ by $w_p(T_1,\ldots,T_n):= \sup_{\| x \| =1} \left(\sum_{i=1}^{n}| \langle T_i x, x \rangle |^p \right)^{\frac1p}$ for $p\geq1$. We generalize some inequalities including Euclidean operator radius of two operators to those involving $w_p$. Further we obtain some lower and upper bounds for $w_p$. Our main result states that if $f$ and $g$ are nonnegative continuous functions on $\left[ 0,\infty \right) $ satisfying $f\left( t\right) g\left(t\right) =t$ for all …

投影体的广义测度扩张

General Measure Extensions of Projection Bodies

本文将 Petty–Zhang 仿射等周型不等式推广到任意测度，并扩张投影体算子至任意测度与函数，建立相应 Zhang 型不等式，最后应用于标准高斯测度。

$L_p$ geominimal surface areas and their inequalities

In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ affine surface area -- a fundamental object in (affine) convex geometry. We prove some properties for the $L_p$ geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and a Santaló style inequality. Cyclic inequalities are established to…

Szarek 不等式与 Ball 不等式的分布稳定性

Distributional stability of the Szarek and Ball inequalities

本文证明 Szarek 最优 Khinchin 不等式与 Ball 立方体切片不等式在接近 Rademacher 分布且权重受 1/√2 分数约束时的分布稳定性，采用傅里叶分析方法。

论Gilles Pisier的高斯集中、等周性与庞加莱型不等式研究方法

On Gilles Pisier's approach to gaussian concentration, isoperimetry and Poincaré-type inequalities

本研究拓展了Gilles Pisier的研究方法，可在欧氏空间旋转不变概率分布（含多维柯西测度）中探索高斯测度相关集中、等周与庞加莱型不等式结果的对应形式。

Extensions of Fiedler-Markham's inequality and Thompson's inequality

We present some new inequalities related to determinant and trace for positive semidefinite block matrices by using symmetric tensor product, which are extensions of Fiedler-Markham's inequality and Thompson's inequality.

粒子梯度下降的误差界限，以及对 log-Sobolev 和 Talagrand 不等式的扩展

Error bounds for particle gradient descent, and extensions of the log-Sobolev and Talagrand inequalities

本文证明了粒子梯度下降（PGD）的非渐近误差界限，通过扩展 log-Sobolev 和 Polyak--Łojasiewicz 不等式及 Bakry--Émery 定理，证明了在强凹对数似然模型下该流指数收敛并控制了离散化误差。

通过集中不等式估计最优传输映射的增长

Growth estimates on optimal transport maps via concentration inequalities

本文通过集中不等式和映射单调性，为Brenier映射的多项式上界提供替代证明并扩展结果，涵盖从高斯到对数凹测度的二次增长情形。