Some Bombieri Type Inequalities in Inner Product

Advances on Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner Product Spaces

The purpose of this survey is to give a comprehensive introduction to some classes of classical and recent analytic inequalities in Inner Product Spaces.

Bessel Type Inequalities for Non-Orthonormal Families of Vectors in Inner Product Spaces

Some sharp Bessel type inequalities for non-orthonormal families of vectors in inner product spaces are given. Applications for complex numbers are also provided.

A Generalisation of the Cassels and Greub-Reinboldt Inequalities in Inner Product Spaces

A generalisation of the Cassels and Greub-Reinboldt inequalities in complex or real inner product spaces and applications for isotonic linear functionals, integrals and sequences are provided.

New Reverses of Schwarz, Triangle and Bessel Inequalities in Inner Product Spaces

New reverses of the Schwarz, triangle and Bessel inequalities in inner product spaces are pointed out. These results complement the recent ones obtained by the author in an earlier paper. Further, they are employed to establish new Gruss type inequalities. Finally, some natural integral inequalities are stated as well.

Some New Results Related to Bessel and Gruss Inequalities for Orthogonal Families in Inner Product Spaces

Some new counterparts of Bessel's inequality for orthornormal families in real or complex inner product spaces are pointed out. Applications for some Gruss type inequalities are also empahsized.

On Bessel and Gruss Inequalities for Orthornormal Families in Inner Product Spaces

A new counterpart of Bessel's inequality for orthornormal families in real or complex inner product spaces is obtained. Applications for some Gruss type results 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.

Some Gruss Type Inequalities in Inner Product Spaces

Some new Gruss type inequalities in inner product spaces and applications for integrals are given.

A Counterpart of Bessel's Inequality in Inner Product Spaces and Some Gruss Type Related Results

A counterpart of the famous Bessel's inequality for orthornormal families in real or complex inner product spaces is given. Applications for some Gruss type inequalities are also provided.

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.

内积空间中 Gruss 型离散不等式再探

Gruss Type Discrete Inequalities in Inner Product Spaces, Revisited

本文重新探讨内积空间中的 Gruss 型离散不等式，得到尖锐结果并应用于凸函数的 Jensen 不等式。

Landau and Gruss type inequalities for inner product type integral transformers in norm ideals

For a probability measure $μ$ and for square integrable fields $(\mathscr{A}_t)$ and $(\mathscr{B}_t)$ ($t\inΩ$) of commuting normal operators we prove Landau type inequality \llu\int_Ω\mathscr{A}_tX\mathscr{B}_tdμ(t)- \int_Ω\mathscr{A}_t\,dμ(t)X \int_Ω\mathscr{B}_t\,dμ(t) \rru \le \llu \sqrt{\,\int_Ω|\mathscr{A}_t|^2\dt-|\int_Ω\mathscr{A}_t\dt|^2}X \sqrt{\,\int_Ω|\mathscr{B}_t|^2 \dt-|\int_Ω\mathscr{B}_t\dt|^2} \rru for all $X\in\mathcalb{B}(\mathcal{H})$ and for all unitarily invariant norms $\lluo\cdot\rruo$. For Schatten $p$-norms similar inequalities are given for arbitrary double square …

两种Hornich-Hlawka型及Gram矩阵-based不等式

Two Hornich-Hlawka-type and Gram matrix-based inequalities

本文在实内积空间中建立两种不等式，第一种是对经典Hornich-Hlawka不等式的乘法加强，探讨了等号成立条件；第二种基于Gram矩阵的半正定性，推导出参数不等式，并通过优化参数得到与Gram行列式不等式相关的紧致不等式。

加强的Cauchy-Schwarz和Hölder不等式

Strengthened Cauchy-Schwarz and Hölder inequalities

本文通过推导恒等式，给出了复内积空间中加强Cauchy-Schwarz不等式基本结果的新证明，并推广到加强Hölder不等式。

Two Weight Inequalities for the Cauchy Transform from $ \mathbb{R}$ to $ \mathbb{C}_+$

We characterize those pairs of weights $ σ$ on $ \mathbb{R}$ and $ τ$ on $ \mathbb{C}_+$ for which the Cauchy transform $\mathsf{C}_σ f (z) \equiv \int_{\mathbb{R}} \frac {f(x)} {x-z} \; σ(dx)$, $ z\in \mathbb{C}_+$, is bounded from $L ^2(\mathbb{R};σ)$ to $L ^{2}(\mathbb{C}_+; τ)$. The characterization is in terms of an $A_2$ condition on the pair of weights and testing conditions for the transform, extending the recent solution of the two weight inequality for the Hilbert transform. As corollaries of this result we derive (1) a characterization of embedding measures for the model space $K_\v…

内函数作为强极点：稳定性性质

Inner functions as strongly extreme points: stability properties

本文研究了H∞空间的有限约束子空间中强极点的稳定性性质，证明了内函数仍为强极点，并探讨了非内函数与强极点的差异。

Inequalities and geometry of hyperbolic-type metrics, radius problems and norm estimates

We consider certain inequalities among the Apollonian metric, the Apollonian inner metric, the $j$ metric and the quasihyperbolic metric. We verify that whether these inequalities can occur in simply connected planar domains and in proper subdomains of $\mathbb{R}^n~(n\ge 2)$. We have seen from our verification that most of the cases cannot occur. This means that there are many restrictions on domains in which these inequalities can occur. We also consider two metrics $j$ and $d$, and investigate whether a plane domain $D\varsubsetneq\mathbb{C}$, for which there exists a constant $c>0$ with $j…

Convergence to global equilibrium for Fokker-Planck equations on a graph and Talagrand-type inequalities

In recent work, Chow, Huang, Li and Zhou introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When $N\ge 2$ is the number of vertices of the graph, they show that the corresponding Fokker-Planck equation is a system of $N$ nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. The different choices for inner products on the space of probability distributions result in different Fokker-Planck equations for the same process. Each of these Fokker-Planck equations has a unique global equilibrium, whi…

Dominating Sets in Bergman Spaces on Strongly Pseudoconvex Domains

We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudoconvex domains in terms of a density condition or a testing condition on the reproducing kernels. Our methods also yield a sufficient conditio…