A Discrete Multi-Sequence Cauchy-Schwarz-Like Inequality

Further refinements of the Cauchy-Schwarz inequality for matrices

Let $A, B$ and $X$ be $n\times n$ matrices such that $A, B$ are positive semidefinite. We present some refinements of the matrix Cauchy-Schwarz inequality by using some integration techniques and various refinements of the Hermite--Hadamard inequality. In particular, we establish the inequality \begin{align*} |||\,|A^{1\over2}XB^{1\over2}|^r|||^2&\leq|||\,|A^{t}XB^{1-s}|^r||| \,\,\,|||\,|A^{1-t}XB^{s}|^r|||\\& \leq\max \{|||\,|AX|^r||| \,\,\,|||\,|XB|^r|||,|||\,|AXB|^r||| \,\,\,|||\,|X|^r|||\}, \end{align*} where $s,t\in[0,1]$ and $r\geq0$.

Some Cauchy-Bunyakovsky-Schwarz Type Inequalities for Sequences of Operators in Hilbert Spaces

Some inequalities of Cauchy-Bunyakovsky-Schwarz type for sequences of bounded linear operators in Hilbert spaces and some applications are given.

Some Reverses of the Cauchy-Bunyakovsky-Schwarz Inequality in 2-Inner Product Spaces

In this paper, some reverses of the Cauchy-Bunyakovsky-Schwarz inequality in 2-inner product spaces are given. Using this framework, some applications for determinantal integral inequalities are also provided.

On Cauchy-Schwarz type inequalities and applications to numerical radius inequalities

In this work, a refinement of the Cauchy--Schwarz inequality in inner product space is proved. A more general refinement of the Kato's inequality or the so called mixed Schwarz inequality is established. Refinements of some famous numerical radius inequalities are also pointed out. As shown in this work, these refinements generalize and refine some recent and old results obtained in literature. Among others, it is proved that if $T\in\mathscr{B}\left(\mathscr{H}\right)$, then \begin{align*} ω^{2}\left(T\right) &\le \frac{1}{12} \left\| \left| T \right|+\left| {T^* } \right|\right\|^2 + \frac{1…

Cauchy-Schwarz不等式的改进及其对数值半径不等式的影响

Enhancement of the Cauchy-Schwarz Inequality and Its Implications for Numerical Radius Inequalities

本文改进了Cauchy-Schwarz不等式，并利用该结果推导出数值半径的新上界。

Reverse Cauchy--Schwarz inequalities for positive C*-valued sesquilinear forms

We prove two new reverse Cauchy--Schwarz inequalities of additive and multiplicative types in a space equipped with a positive sesquilinear form with values in a C*-algebra. We apply our results to get some norm and integral inequalities. As a consequence, we improve a celebrated reverse Cauchy--Schwarz inequality due to G Polya and G. Szego.

On the generalized mixed Schwarz inequality

In this work, an extension of the generalized mixed Schwarz inequality is proved. A companion of the generalized mixed Schwarz inequality is established by merging both Cartesian and Polar decompositions of operators. Based on that some numerical radius inequalities are proved.

Improving Schwarz Inequality in Inner Product Spaces

Some improvements of the celebrated Schwarz inequality in complex inner product spaces are given. Applications for n-tuples of complex numbers are provided.

On the Cauchy--Rassias Inequality and Linear n-Inner Product Preserving Mappings

We prove the Cauchy-Rassias stability of linear n-inner product preserving mappings in $n$-inner product Banach spaces. We apply the Cauchy-Rassias inequality that plays an influencial role in the subject of functional equations. The inequality was introduced for the first time by Th.M.Rassias in his paper entitled: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math.Soc. 72(1978), 297-300.

通过矩阵 Cauchy-Schwarz 不等式给出 Lee 关于 Frobenius 范数猜想的新证明

A new proof of Lee's conjecture on the Frobenius norm via the matrix Cauchy-Schwarz inequality

本文仅依靠矩阵 Cauchy-Schwarz 不等式，为 Eun-Young Lee 关于复矩阵 Frobenius 范数的不等式猜想提供了一种新证明。

On the conjecture of the norm Schwarz inequality

For any positive invertible matrix $A$ and any normal matrix $B$ in $M_{n}({\Bbb C})$, we investigate whether the inequality $ ||A\sharp (B^{*}A^{-1}B)||\geq ||B|| $ is true or not, where $\sharp$ denotes the geometric mean and $||\cdot||$ denotes the operator norm. We will solve this problem negatively. The related topics are also discussed.

A treatment of the Cauchy--Schwarz inequality in $C^*$-modules

We study the Cauchy--Schwarz and some related inequalities in a semi-inner product module over a $C^*$-algebra $\A$. The key idea is to consider a semi-inner product $\A$-module as a semi-inner product $\A$-module with respect to another semi-inner product. In this way, we improve some inequalities such as the Ostrowski inequality and an inequality related to the Gram matrix. The induced semi-inner products are also related to the the notion of covariance and variance. Furthermore, we obtain a sequence of nested inequalities that emerges from the Cauchy--Schwarz inequality. As a consequence, w…

关于Cauchy-Schwarz-Bunyakovsky不等式的一般化及其在弹性力学中的应用

On the Generalizations of the Cauchy-Schwarz-Bunyakovsky Inequality with Applications to Elasticity

本文系统阐述了CBS不等式的离散与积分形式、在n维欧几里得空间及其线性子空间中的推广及加强形式，并给出几何解释和关键证明，强调了其在弹性问题中的重要作用。

非交换Lp空间中映射的柯西-施瓦茨不等式

Cauchy-Schwarz inequalities for maps in noncommutative Lp-spaces

本文研究了非交换Lp空间中正半线性映射的广义柯西-施瓦茨不等式，推导了实部和虚部的界估计，并在非交换L2空间中推广了不确定性关系。还提出了非交换L2空间的新范数，并证明了正半线性映射的柯西-施瓦茨不等式。

A sharpened Schwarz-Pick operatorial inequality for nilpotent operators

Let denote by $S(ϕ)$ the extremal operator defined by the compression of the unilateral shift $S$ to the model subspace $ H(ϕ)=H^{2} \ominus ϕH^{2} $ as the following $S(ϕ)f(z)=P(zf(z)),$ where $P$ denotes the orthogonal projection from the Hardy space $H^{2}$ onto $ H(ϕ)$ and $ϕ$ is an inner function on the unit disc. In this mathematical notes, we give an explicit formula of the numerical radius of the truncated shift $S(ϕ)$ in the particular case where $ϕ$ is a finite Blaschke product with unique zero and an estimate on the general case. We establish also a sharpened Schwarz-Pick operatoria…

A Counterpart of Schwarz's Inequality in Inner Product Spaces

A new counterpart of Schwarz's inequality in inner product spaces and applications for isotonic functionals, integrals and sequences are provided.

The applications of Cauchy-Schwartz inequality for Hilbert modules to elementary operators and i.p.t.i. transformers

We apply the inequality $\left|\left<x,y\right>\right|\le\|x\|\,\left<y,y\right>^{1/2}$ to give an easy and elementary proof of many operator inequalities for elementary operators and inner type product integral transformers obtained during last two decades, which also generalizes all of them.

Reverses of the Schwarz Inequality in Inner Product Spaces Generalising a Klamkin-McLenaghan Result

New reverses of the Schwarz inequality in inner product spaces that incorporate the classical Klamkin-McLenaghan result for the case of positive n-tuples are given. Applications for Lebesgue integrals are also provided.

Refinements of Some Reverses of Schwarz's Inequality in 2-Inner Product Spaces and Applications for Integrals

Refinements of some recent reverse inequalities for the celebrated Cauchy-Bunyakovsky-Schwarz inequality in 2-inner product spaces are given. Using this framework, applications for determinantal integral inequalities are also provided.

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。

The generalized Schwarz inequality for semi-Hilbertian space operators and Some $A$-numerical radius inequalities

In this work, the mixed Schwarz inequality for semi-Hilbertian space operators is proved. Namely, for every positive Hilbert space operator $A$. If $f$ and $g$ are nonnegative continuous functions on $\left[0,\infty\right)$ satisfying $f(t)g(t) =t$ $(t\ge0)$, then \begin{align*} \left| {\left\langle {T x,y} \right\rangle_A } \right| \le \left\| {f\left( {\left| T \right|_A x} \right)} \right\|_A \left\| {g\left( {\left| {T^{\sharp_A } } \right|_A y} \right)} \right\|_A \end{align*} for every Hilbert space operator $T$ such that the range of $T^* A$ is a subset in the range of $A$, such that $A…

Schwarz-Pick不等式与von Neumann不等式的若干关系

Some relations between Schwarz-Pick inequality and von Neumann's inequality

该研究借助算子理论方法，利用2×2矩阵组的von Neumann不等式推导Schur-Agler类的Schwarz-Pick型不等式，给出完全谱域的充分条件并推广相关von Neumann不等式。

有限点的Schwarz-Pick不等式

A finite-Point Schwarz-Pick Inequality

该研究证明了圆盘全纯自映射的有限点版本Schwarz-Pick不等式，通过比较Bergman Gram矩阵得到精确常数，其取值与点数量及有限Blaschke乘积的映射次数相关。

双圆盘上不定Schwarz-Pick不等式的RKHS方法

An RKHS approach to the indefinite Schwarz-Pick inequality on the bidisk

本文利用复几何与再生核Hilbert空间几何的联系，推广了Seto关于双圆盘上不定Schwarz-Pick不等式的结果。

Operator versions of Hölder inequality and Hilbert $C^*$-modules

Recently proved weighted Cauchy Scwarz inequality for Hilbert $C^*$-modules leads to many Hölder type inequalities for unitarily invariant norms on Hilbert space operators.

Gram矩阵的Hadamard幂与广义柯西-施瓦茨不等式

Hadamard powers of Gram matrices and Generalized Cauchy-Schwarz inequalities

该论文证明了Gram矩阵Hadamard幂相关的两个猜想，确定不等式取等条件，还提出其针对三个及以上向量的新推广。

推广柯西-施瓦茨不等式：Hadamard幂与张量积

Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products

该研究推广了一类柯西-施瓦茨型不等式，给出三种新阐释性证明，并从向量到矩阵、非2指数、多向量三个方向拓展了该不等式的适用范围。

Self-mappings of the quaternionic unit ball: multiplier properties, Schwarz-Pick inequality, and Nevanlinna--Pick interpolation problem

We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space. In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.

必然性论述与 Cauchy-Bunyakovsky-Schwarz 不等式

Apodictic discourse and the Cauchy-Bunyakovsky-Schwarz inequality

本文从历史与论述结构视角探讨 Bunyakovsky 积分不等式的起源、动机及后续数学家（Liouville、Grassmann、Schwarz 等）的重新解释，揭示了该不等式在概率统计与积分方程理论中的演变与推广。

Norm inequalities related to the Heron and Heinz means

In this article, we present several inequalities treating operator means and the Cauchy-Schwarz inequality. In particular, we present some new comparisons between operator Heron and Heinz means, several generalizations of the difference version of the Heinz means and further refinements of the Cauchy-Schwarz inequality. The techniques used to accomplish these results include convexity and Löwner matrices.

从算子理论视角看Schwarz-Pick型不等式

Schwarz-Pick type inequalities from an operator theoretical point of view

从算子理论视角，利用von Neumann不等式证明并推广了Schwarz-Pick型不等式，包括多点情形、算子版本及多圆盘与高阶导数情形，并借助判别簇增强了双圆盘上的Schwarz-Pick引理。

Characterizing Schwarz maps by tracial inequalities

Let $ϕ$ be a linear map from the $n\times n$ matrices ${\mathcal M}_n$ to the $m\times m$ matrices ${\mathcal M}_m$. It is known that $ϕ$ is $2$-positive if and only if for all $K\in {\mathcal M}_n$ and all strictly positive $X\in {\mathcal M}_n$, $ϕ(K^*X^{-1}K) \geq ϕ(K)^*ϕ(X)^{-1}ϕ(K)$. This inequality is not generally true if $ϕ$ is merely a Schwarz map. We show that the corresponding tracial inequality ${\rm Tr}[ϕ(K^*X^{-1}K)] \geq {\rm Tr}[ϕ(K)^*ϕ(X)^{-1}ϕ(K)]$ holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with vario…