Sever Silvestru Dragomir
Some elementary inequalities providing upper bounds for the difference of the norm and the numerical radius of a bounded linear operator on Hilbert spaces under appropriate conditions are given.
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Some rearrangement inequalities for symmetric norms on matrices are given as well as related results for operator convex functions.
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.
Some inequalities of Cauchy-Bunyakovsky-Schwarz type for sequences of bounded linear operators in Hilbert spaces and some applications are given.
This article is devoted to studying some new numerical radius inequalities for Hilbert space operators. Our analysis enables us to improve an earlier bound of numerical radius due to Kittaneh.
Some new inequalities for the norm and the numerical radius of composite operators generated by a pair of operators are given.
The main aim of this book is to present recent results concerning inequalities for continuous functions of selfadjoint operators on complex Hilbert spaces. It is intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas.
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.
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。
An operator $T$ acting on a Hilbert space is called $(α,β)$-normal ($0\leq α\leq 1\leq β$) if \begin{equation*} α^{2}T^{\ast }T\leq TT^{\ast}\leq β^{2}T^{\ast}T. \end{equation*} In this paper we establish various inequalities between the operator norm and its numerical radius of $(α,β)$-normal operators in Hilbert spaces. For this purpose, we employ some classical inequalities for vectors in inner product spaces.
In this paper, we give some reverse-types of Ando's and Hölder-McCarthy's inequalities for positive linear maps, and positive invertible operators. For our purpose, we use a recently improved Young inequality and its reverse.
Some reverse inequalities for scalar Birkhoff weak integrable functions
本文研究了标量Birkhoff弱可积函数在非加法测度下的Hölder和Minkowski不等式及其反向不等式。
In this paper, we give a new inequality for convex functions of real variables, and we apply this inequality to obtain considerable generalizations, refinements, and reverses of the Young and Heinz inequalities for positive scalars. Applications to unitarily invariant norm inequalities involving positive semidefinite matrices are also given.
Regarding the generalizations of the Bessel inequality in Hilbert spaces which are due to Bombiari and Boas--Bellman, we obtain a version of the Bessel inequality and some generalizations of this inequality in the framework of Hilbert $C^*$-modules.
In this paper, we show refined Young inequalities for two positive operators. Our results refine the ordering relations among the arithmetic mean, the geometric mean and the harmonic mean for two positive operators. In addition, we give two different reverse inequalities for the refined Young inequality for two positive operators.
We examine a number of known inequalities for $L^p$ functions with reverse representations for $s<1$ with complex matrices under the $p$-norms $||X||_p=\text{Tr}[(X^\ast X)^{p/2}]^{1/p}$, and similarly defined quasinorm or antinorm quantities $||X||_s=\text{Tr}[(X^\ast X)^{s/2}]^{1/s}$. Analogous to the reverse Hölder and reverse Minkowski for $L^p$ functions, it has recently been shown that for $A,B\in M_{n\times n}(\mathbb{C})$ such that $|B|$ is invertible, $||AB||_1\geq ||A||_{s}||B||_{s/(s-1)}$ and for $A,B$ positive semidefinite that $||A+B||_s\geq ||A||_s+||B||_s$. We comment on variati…
Refining some results of S. S. Dragomir, several new reverses of the triangle inequality in inner product spaces are obtained.
We provide a precise statement and self contained proof of a Sobolev inequality (cf. [A, page 236 and page 237]) stated in the original paper. Higher order and fractional inequalities are treated as well.
In this paper, more inequalities between the operator norm and its numerical radius, for the class of normal operators, are established. Some of the obtained results are based on recent reverse results for the Schwarz inequality in Hilbert spaces due to the author.
Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces that improve some earlier results are pointed out. They are applied to obtain new Gruss type inequalities in inner product spaces. Some natural applications for integral inequalities are also pointed out.
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.
Two reverse inequalities for Young's inequality were shown by M. Tominaga, using Specht ratio. In this short paper, we show alternative reverse inequalities for Young's inequality without using Specht ratio.
A new approach to classical self improving results for $BMO$ functions is presented. "Coordinate Gagliardo spaces" are introduced and a generalized version of the John-Nirenberg Lemma is proved. Applications are provided.
Extending results in \cite{M} and \cite{MM} we characterize the classical classes of weights that satisfy reverse Hölder inequalities in terms of indices of suitable families of $K-$functionals of the weights. In particular, we introduce a Samko type of index (cf. \cite{kara}) for families of functions, that is based on quasi-monotonicity, and use it to provide an index characterization of the $RH_{p}$ classes, as well as the limiting class $RH=$ $RH_{LLogL}=$. $\bigcup\limits_{p>1}RH_{p}$ (cf. \cite{BMR}),\ which in the abstract case involves extrapolation spaces. Reverse Hölder inequalities …
We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak B}(\mathcal{H})$ are positive operators and $r\geq 0$, $A\nabla_{-r}B+2r(A\nabla B-A\sharp B)\leq A\sharp_{-r}B$ and prove that equality holds if and only if $A=B$. We also establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if $A, B$ are positive definite matrices and $r\geq 0$, then …
Discrete reversed Hardy inequalities and Hölder estimates in Lorentz sequence spaces
本文在洛伦茨序列空间中建立了新的最优反 Hardy 型不等式及不同范数间的估计，推导出补充函数情况的尖锐霍尔德型不等式。
In recent years, it has been shown that some classical inequalities follow from a local stochastic dominance for naturally associated random polytopes. We strengthen planar isoperimetric inequalities by attaching a stochastic model to some classical inequalities, such as Mahler's Theorem, and a reverse Lutwak-Zhang inequality, the polar for $L_p$ centroid bodies. In particular, we obtain the dual counterpart to a result of Bisztriczky-Böröczky.
We prove several singular value inequalities for sum and product of compact operators in Hilbert space. Some of our results generalize the previous inequalities for operators. Also, applications of some inequalities are given.
We present more than 50 results including some range inclusion results to characterize reverse order law for Moore-Penrose inverse of closed range Hilbert space operators. We use basic properties of Moore-Penrose inverse to prove the results. Some examples are also provided to illustrate failure cases to hold the reverse order law in infinite dimensional settings.
In this work, some generalizations and refinements inequalities for numerical radius of the product of Hilbert space operators are proved. New inequalities for numerical radius of block matrices of Hilbert space operators are also established.
In this paper we present some inequalities involving operator decreasing functions and operator means. These inequalities provide some reverses of operator Aczél inequality dealing with the weighted geometric mean.
We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be seen as a version of Milman's reverse Brunn-Minkowski inequality. The proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies, and on a study of effective supports of convex measures, both of which are of independent interest, as well as on Milman's deep technology of $M$-ellip…
Reverse Hölder inequalities for a class of functions on a probability space constitute an important tool in Analysis in Probability. After revisiting how a (modified) log-Sobolev inequality can be used to derive reverse Hölder inequalities for the class of log-Lipschitz functions, we obtain a weaker condition using general Transport-Entropy inequalities, which can also handle approximately log-Lipschitz functions. In its weakest form, the condition degenerates to the assumption of satisfying a concentration inequality. We compare this with a scenario in which the underlying space only satisfie…
Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals
本文在对偶Brunn-Minkowski理论中建立了两个新几何不等式：证明了Lutwak猜想的原点对称凸体对偶均质积分的Brunn-Minkowski不等式，并推广Ball体积比不等式得出在John位置下立方体最大化对偶均质积分的逆向等周不等式。
We prove several versions of reverse triangle inequality in Hilbert $C^*$-modules. We show that if $e_1, ..., e_m$ are vectors in a Hilbert module ${\mathfrak X}$ over a $C^*$-algebra ${\mathfrak A}$ with unit 1 such that $<e_i,e_j>=0 (1\leq i\neq j \leq m)$ and $\|e_i\|=1 (1\leq i\leq m)$, and also $r_k,ρ_k\in\Bbb{R} (1\leq k\leq m)$ and $x_1, ..., x_n\in {\mathfrak X}$ satisfy $$0\leq r_k^2 \|x_j\|\leq {Re}< r_ke_k,x_j> ,\quad0\leq ρ_k^2 \|x_j\| \leq {Im}< ρ_ke_k,x_j> ,$$ then [\sum_{k=1}^m(r_k^2+ρ_k^2)]^{1/2}\sum_{j=1}^n \|x_j\|\leq\|\sum_{j=1}^nx_j\|, and the equality holds if and only if …
In this work, generalization of some inequalities for continuous $h$-synchronous ($h$-asynchronous) functions of selfadjoint linear operators in Hilbert spaces are proved.
Some new trace inequalities for operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated and applications for power series of such operators are given. Some trace inequalities for matrices are also derived. Examples for the operator exponential and other similar functions are presented as well.
Inequalities for the $A$-Norm and $A$-Numerical Radius of Operator Sums in Semi-Hilbertian Spaces with Applications
本文对半希尔伯特空间中算子和的$A$-范数与$A$-数值半径建立新不等式，改进广义三角不等式并给出算子乘积与交换子的新边界，可应用于量子力学和偏微分方程，证明了不等式的最优性与优势。
For a bounded linear operator $A$ on a reproducing kernel Hilbert space $\mathcal{H}(Ω)$, with normalized reproducing kernel $\widehat{k}_λ = \frac{k_λ}{\lVert k_λ\lVert}$, the Berezin symbol, Berezin number and Berezin norm are defined respectively by $\widetilde{A}(λ) = \langle A\widehat{k}_λ,\widehat{k}_λ\rangle$, $ber(A) = \sup_{λ\inΩ}\left|\widetilde{A}(λ)\right|$ and $\left\|A\right\|_{ber} = \sup_{λ\inΩ}\left\|A\widehat{k}_λ\right\|$. A straightforward comparison between these characteristics yields the inequalities $ber(A)\leq\left\|A\right\|_{ber}\leq\lVert A\lVert$. In this paper, we…
Some reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in Hilbert spaces are given. Applications for complex-valued functions are provided as well.
In this paper we obtain some operator versions of Levin-Steckin integral inequality.
In this note, some refinements of Young inequality and its reverse for positive numbers are proved and using these inequalities some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained.
Some additive reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in Hilbert spaces are given. Applications for complex-valued functions are provided as well.
Some quadratic reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in Hilbert spaces are given. Applications for complex-valued functions are provided as well.
Block Positivity and Optimal Mixed-Schwarz Inequalities on Hilbert $C^*$-Modules
本文研究Hilbert $C^*$-模上可伴随算子理论，给出$n\ imes n$块算子矩阵正性的等价可验证条件，推导紧的带参数混合施瓦茨不等式及极值算子刻画，二者互为支撑并给出相关应用。
We prove that for any weight $ϕ$ defined on $[0,1]^n$ that satisfies a reverse Holder inequality with exponent p > 1 and constant $c\ge1$ upon all dyadic subcubes of $[0,1]^n$, it's non increasing rearrangement satisfies a reverse Holder inequality with the same exponent and constant not more than $2^nc-2^n + 1$, upon all subintervals of $[0; 1]$ of the form $[0; t]$. This gives as a consequence, according to the results in [8], an interval $[p; p_0(p; c)) = I{p,c}$, such that for any $q \in I{p,c}$, we have that $ϕ$ is in $L^q$.
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.
This paper presents a number of Kantorovich type integral inequalities involving tensor products of continuous fields of bounded linear operators on a Hilbert space. Kantorovich type inequality in which the product is replaced by an operator mean is also considered. Such inequalities include discrete inequalities as special cases. Moreover, some generalizations of an additive Gruss integral inequality for operators are obtained.
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。
In this paper, we present some operator and eigenvalue inequalities involving operator monotone, doubly concave and doubly convex functions. These inequalities provide some variants of operator Aczél inequality and its reverse via generalized Kantorovich constant.
Given a centered convex body $K\subseteq\mathbb{R}^n$, we study the optimal value of the constant $\tildeΛ(K)$ such that there exists an orthonormal basis $\{w_i\}_{i=1}^n$ for which the following reverse dual Loomis-Whitney inequality holds: $$ |K|^{n-1}\leqslant \tildeΛ(K)\prod_{i=1}^n|K\cap w_i^\perp|. $$ We prove that $\tildeΛ(K)\leqslant(CL_K)^n$ for some absolute $C>1$ and that this estimate in terms of $L_K$, the isotropic constant of $K$, is asymptotically sharp in the sense that there exists another absolute constant $c>1$ and a convex body $K$ such that $(cL_K)^n\leqslant\tildeΛ(K)\l…
In this paper we establish Parseval type identities and surprising new inequalities for Hilbert-Schmidt frames. Our results generalize and improve the remarkable results which have been obtained by Balan et al. and G{ă}vru{ţ}a.
Convex inequalities in Hilbert $C^*$-modules
本文将H$"o$lder-McCarty不等式中的凸不等式推广至Hilbert $C^*$-模，并研究该情形下的相关不等式。
In this paper we characterize the validity of the inequalities $\|g\|_{p,(a,b),λ} \le c \|u(x) \|g\|_{\infty,(x,b),μ}\|_{q,(a,b),ν}$ and $\label{eq.0.1.2} \|g\|_{p,(a,b),λ} \le c \|u(x) \|g\|_{\infty,(a,x),μ}\|_{q,(a,b),ν} $ for all non-negative Borel measurable functions $g$ on the interval $(a,b)$, where $0 < p \le +\infty$, $0 < q \le +\infty$, $λ$, $μ$ and $ν$ are non-negative Borel measures on $(a,b)$, and $u$ is a weight function on $(a,b)$.
We generalize several inequalities involving powers of the numerical radius for product of two operators acting on a Hilbert space. For any $A, B, X\in \mathbb{B}(\mathscr{H})$ such that $A,B$ are positive, we establish some numerical radius inequalities for $A^αXB^α$ and $A^αX B^{1-α}\,\,(0 \leq α\leq 1)$ and Heinz means under mild conditions.
In this paper we present some reverses of the Golden-Thompson type inequalities: Let $H$ and $K$ be Hermitian matrices such that $ e^s e^H \preceq_{ols} e^K \preceq_{ols} e^t e^H$ for some scalars $s \leq t$, and $α\in [0 , 1]$. Then for all $p>0$ and $k =1,2,\ldots, n$ \begin{align*} \label{} λ_k (e^{(1-α)H + αK} ) \leq (\max \lbrace S(e^{sp}), S(e^{tp})\rbrace)^{\frac{1}{p}} λ_k (e^{pH} \sharp_αe^{pK})^{\frac{1}{p}}, \end{align*} where $A\sharp_αB = A^\frac{1}{2} \big ( A^{-\frac{1}{2}} B^\frac{1}{2} A^{-\frac{1}{2}} \big) ^αA^\frac{1}{2}$ is $α$-geometric mean, $S(t)$ is the so called Spech…
In this paper, we establish Parseval identities and surprising new inequalities for weaving frames in Hilbert space, which involve scalar $λ\in\rs$. By suitable choices of $λ$, one obtains the previous results as special cases. Our results generalize and improve the remarkable results which have been obtained by Balan et al. and Găvruţa.
Hardy-Hilbert type inequalities on homogeneous groups-An introduction and generalization to the kernel case
本文介绍齐次群上Hardy-Hilbert型不等式的发展，将经典情形的若干该类不等式统一为一般核情形，并把结果推广至齐次群情形。
Indefinite Schwarz-Pick inequalities for holomorphic self-maps of the bidisk are given as application of the spectral theory on Hilbert modules.
From simplex slicing to sharp reverse Hölder inequalities
本文将单纯形切片理论扩展到概率框架，推导了针对中心对称对数凹随机变量的尖锐反霍尔德不等式，并揭示了极值分布的相变现象。
Some inequalities for the Euclidean operator radius of two operators in Hilbert $C^{\ast}$-Modules space
本文研究Hilbert $C^*$-模中两个有界线性算子的欧几里得算子半径，给出若干精确界，并将其与已有数值半径上下界结果建立联系。
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