Quadratic Reverses of the Triangle Inequality in Inner Product Spaces

Quadratic Reverses of the Continuous Triangle Inequality for Bochner Integral of Vector-Valued Functions in Hilbert Spaces

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.

Some Reverses of the Continuous Triangle Inequality for Bochner Integral of Vector-Valued Functions in Complex Hilbert Spaces

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.

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

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.

Reverses of the Triangle Inequality in Banach Spaces

Recent reverses for the discrete generalised triangle inequality and its continuous version for vector-valued integrals in Banach spaces are surveyed. New results are also obtained. Particular instances of interest in Hilbert spaces and for complex numbers and functions are pointed out as well.

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.

Inequalities that sharpen the triangle inequality for sums of $N$ functions in $L^p$

We study $L^p$ inequalities that sharpen the triangle inequality for sums of $N$ functions in $L^p$.

Operator inequalities via the triangle inequality

This article improves the triangle inequality for complex numbers, using the Hermite-Hadamard inequality for convex functions. Then, applications of the obtained refinement are presented to include some operator inequalities. The operator applications include numerical radius inequalities and operator mean inequalities.

More on Reverse Triangle Inequality in Inner Product Spaces

Refining some results of S. S. Dragomir, several new reverses of the generalized triangle inequality in inner product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if $a$ is a unit vector in a real or complex inner product space $(H;< .,.>)$, $r, s>0, p\in(0,s], D=\{x\in H,\|rx-sa\|\leq p\}, x_1, x_2\in D-\{0\}$ and $ α_{r,s}=\min\{\frac{r^2\|x_k\|^2-p^2+s^2}{2rs\|x_k\|}: 1\leq k\leq 2 \}$, then $$\frac{\|x_1\|\|x_2\|-Re< x_1,x_2>}{(\|x_1\|+\|x_2\|)^2}\leq α_{r,s}.$$

关于曲率二次衰减流形上的Riesz变换及其反向不等式

On the Riesz transform and its reverse inequality on manifolds with quadratically decaying curvature

本文研究了曲率二次衰减流形上的Riesz变换及其反向不等式，通过Lorentz型估计和谐波环消方法，证明了反向Riesz、Hardy和加权Sobolev不等式等价，并验证了Grushin空间中Riesz变换的有界性。

On a reverse form of the Brascamp-Lieb inequality

We prove a reverse form of the multidimensional Brascamp-Lieb inequality. Our method also gives a new way to derive the Brascamp-Lieb inequality and is rather convenient for the study of equality cases.

On New Refinements and Reverses of Young's Operator Inequality

In this paper we obtain some new refinements and reverses of Young's operator inequality. Extensions for convex functions of operators are also provided.

Reverses of operator Aczél inequality

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.

Sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb R_+^n$

This is the second in our series of papers concerning some reversed Hardy--Littlewood--Sobolev inequalities. In the present work, we establish the following sharp reversed Hardy--Littlewood--Sobolev inequality on the half space $\mathbb R_+^n$ \[ \int_{\mathbb R_+^n} \int_{\partial \mathbb R_+^n} f(x) |x-y|^λg(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p(\partial \mathbb R_+^n)} \, \|g\|_{L^r(\mathbb R_+^n)} \] for any nonnegative functions $f\in L^p(\partial \mathbb R_+^n)$, $g\in L^r(\mathbb R_+^n)$, and $p,r\in (0,1)$, $λ> 0$ such that $(1-1/n)1/p + 1/r -(λ-1) /n =2$. Some estimates for…

Some Results About Reverses of Cauchy-Schwarz Inequality in Inner Product Spaces

Some new reverses of the Cauchy-Schwarz inequality in inner product spaces are presented in this paper.The results obtained here complement the recent work of the references.

Sharp reversed Hardy-Littlewood-Sobolev inequality on $\mathbb R^n$

This is the first in our series of papers concerning some Hardy-Littlewood-Sobolev type inequalities. In the present paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space $\mathbb R^n$ \[\int_{\mathbb R^n} \int_{\mathbb R^n} f(x) |x-y|^λg(y) dx dy \geqslant \mathscr C_{n,p,r} \|f\|_{L^p (\mathbb R^n)}\, \|g\|_{L^r (\mathbb R^n)}\] for any nonnegative functions $f\in L^p(\mathbb R^n)$, $g\in L^r(\mathbb R^n)$, and $p,r\in (0,1)$, $λ> 0$ such that $1/p + 1/r -λ/n =2$. We will also explore some estimates for $\mathscr C_{n,p,r}$ and the existence…

On refined operator version of Young inequality and its reverse

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.

关于二次对称模的算子三角不等式

An operator triangle inequality for the quadratic symmetric modulus

本文在50年后重新建立了关于二次对称模的算子三角不等式，并讨论了等式情况和无限维情形，同时获得了Clarkson-McCarthy型不等式，回答了Bourin和Lee提出的问题。

On the operator Aczél inequality and its reverse

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.

量子瓦瑟斯坦距离的三角不等式

Triangle Inequality for a Quantum Wasserstein Divergence

本文通过复分析方法证明了量子2-瓦瑟斯坦距离的三角不等式，解决了相关猜想。

A Hardy inequality and applications to reverse Holder inequalities for weights on $R$

We prove a sharp integral inequality valid for non-negative functions defined on $[0,1]$, with given $L^1$ norm. This is in fact a generalization of the well known integral Hardy inequality. We prove it as a consequence of the respective weighted discrete analogue inequality which proof is presented in this paper. As an application we find the exact best possible range of $p>q$ such that any non-increasing $f$ which satisfies a reverse Hölder inequality with exponent $q$ and constant $c$ upon the subintervals of $[0,1]$, should additionally satisfy a reverse Hölder inequality with exponent $p$…

On a reverse of the Tan-Xie inequality for sector matrices and its applications

In this short paper, we establish a reverse of the derived inequalities for sector matrices by Tan and Xie, with Kantorovich constant. Then, as application of our main theorem, some inequalities for determinant and unitarily invariant norm are presented.

Characterization of a generalized triangle inequality in normed spaces

For a normed linear space $(X,|\cdot|)$ and $p>0$ we characterize all $n$-tuples $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the generalized triangle inequality of the second type $$\|x_1+...+x_n\|^p\leq\frac{|x_1|^p}{μ_1}+...+\frac{|x_n|^p}{μ_n}$$ holds for any $x_1,...,x_n\in X$. We also characterize $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the reverse of the inequality above holds.

Alternative reverse inequalities for Young's inequality

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.

Sharpening the triangle inequality: envelopes between $L^{2}$ and $L^{p}$ spaces

Motivated by the inequality $\|f+g\|_{2}^{2} \leq \|f\|_{2}^{2}+2\|fg\|_{1}+\|g\|^{2}_{2}$, Carbery (2006) raised the question what is the "right" analogue of this estimate in $L^{p}$ for $p \neq 2$. Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an $L^{p}$ version of this inequality by providing upper bounds for $\|f+g\|_{p}^{p}$ in terms of the quantities $\|f\|_{p}^{p}, \|g\|_{p}^{p}$ and $\|fg\|_{p/2}^{p/2}$ when $p \in(0,1] \cup [2,\infty)$, and lower bounds when $p \in (-\infty,0) \cup (1,2)$, thereby proving (and improving) the suggested possible inequalities of Carbery. We…

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.

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.

Reverses and Refinements of Jensen's Inequality for Positive Linear Functionals on Hermitian Unital Banach *-Algebras

We establish in this paper some inequalities for analytic and convex functions on an open interval and positive normalized functionals defined on a Hermitian unital Banach *-algebra. Reverses and refinements of Jensen's and Slater's type inequalities are provided. Some examples for particular convex functions of interest are given as well.

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.

A note on Santaló inequality for the polarity transform and its reverse

We prove a Santaló and a reverse Santaló inequality for the polarity transform, which was recently re-discovered by Artstein-Avidan and Milman, in the class consisting of (even) log-concave functions attaining their maximal value 1 at the origin, also called geometric log-cancave functions. The bounds are sharp up to the optimal universal constants.

A dimension-free reverse logarithmic Sobolev inequality for low-complexity functions in Gaussian space

We discuss new proofs, and new forms, of a reverse logarithmic Sobolev inequality, with respect to the standard Gaussian measure, for low complexity functions, measured in terms of Gaussian-width. In particular, we provide a dimension-free improvement for a related result given in [Eldan '18].

Inequalities for $L^p$-norms that sharpen the triangle inequality and complement Hanner's Inequality

In 2006 Carbery raised a question about an improvement on the naïve norm inequality $\|f+g\|_p^p \leq 2^{p-1}(\|f\|_p^p + \|g\|_p^p)$ for two functions in $L^p$ of any measure space. When $f=g$ this is an equality, but when the supports of $f$ and $g$ are disjoint the factor $2^{p-1}$ is not needed. Carbery's question concerns a proposed interpolation between the two situations for $p>2$. The interpolation parameter measuring the overlap is $\|fg\|_{p/2}$. We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all $…

Reverse triangle inequality in Hilbert $C^*$-modules

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 …

Stability of the reverse Blaschke-Santalo inequality for unconditional convex bodies

Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in R^n. The corresponding inequality to the conjecture is sometimes called the the reverse Blaschke-Santalo inequality. The conjecture is known in dimension two and in several special cases. In the class of unconditional convex bodies, Saint Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, w…

Sharp constants related to the triangle inequality in Lorentz spaces

We study the Lorentz spaces $L^{p,s}(R,μ)$ in the range $1<p<s\le \infty$, for which the standard functional $$ ||f||_{p,s}=(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t})^{1/s} $$ is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: $$ ||f||_{(p,s)}=\inf\bigg\{\sum_{k}||f_k||_{p,s}\bigg\}, $$ where the infimum is taken over all finite representations $f=\sum_{k}f_k. $ We also prove that the decomposition norm and the dual norm $$ ||f||_{p,s}'= \sup\left\{\int_R fg dμ: ||g||_{p',s'}=1\right…

关于范数之和的某个不等式及反向不确定关系

On a certain inequality for the sum of norms and reverse uncertainty relations

本文证明了内积空间中两个向量范数平方和的一个简单不等式，并利用该不等式推导了所谓的“反向不确定关系”并分析其性质。

Reverse Hölder's inequality for spherical harmonics

This paper determines the sharp asymptotic order of the following reverse Hölder inequality for spherical harmonics $Y_n$ of degree $n$ on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$ as $n\to \infty$: \[\|Y_n\|_{L^q(\mathbb{S}^{d-1})}\leq C n^{α(p,q)}\|Y_n\|_{L^p(\mathbb{S}^{d-1})},\quad 0<p<q\leq \infty.\] In many cases, these sharp estimates turn out to be significantly better than the corresponding estimates in the Nilkolskii inequality for spherical polynomials. Furthermore, they allow us to improve two recent results on the restriction conjecture and the sharp Pitt inequalities f…

非线性积分三角不等式的一个刻画

A characterization of a nonlinear integral triangle inequality

本文在 Banach 空间与 Lebesgue 测度空间上，刻画了使非线性积分三角不等式及其反向不等式成立的可测函数 u，并解决了离散对应问题。

Further generalizations, refinements, and reverses of the Young and Heinz inequalities

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.

Volume ratios and a reverse isoperimetric inequality

It is shown that if $C$ is an $n$-dimensional convex body then there is an affine image $\widetilde C$ of $C$ for which $${|\partial \widetilde C|\over |\widetilde C|^{n-1\over n}}$$ is no larger than the corresponding expression for a regular $n$-dimensional ``tetrahedron''. It is also shown that among $n$-dimensional subspaces of $L_p$ (for each $p\in [1,\infty]), \ell^n_p$ has maximal volume ratio.\vskip3in

Refinements of a reversed AM-GM operator inequality

We prove some refinements of a reverse AM-GM operator inequality due to M. Lin [Studia Math. 2013;215:187-194]. In particular, we show the operator inequality \begin{eqnarray*} Φ^p\left(A\nabla_νB+2rMm(A^{-1}\nabla B^{-1}-A^{-1}\sharp B^{-1})\right)\leqα^pΦ^p\left(A\sharp_νB\right), \end{eqnarray*} where $A,B$ are positive operators on a Hilbert space such that $0<m \leq A, B \leq M$ for some positive numbers $m, M$, $Φ$ is a positive unital linear map, $ν\in[0,1]$, $r=\min\{ν,1-ν\}$, $p>0$ and $α=\max\left\{\frac{(M+m)^2}{4Mm},\frac{(M+m)^2}{4^\frac{2}{p}Mm}\right\}$.

Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures

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…

对偶均质积分的Brunn-Minkowski不等式与逆向等周不等式

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。

The case of equality in geometric instances of Barthe's reverse Brascamp-Lieb inequalit

The works of Bennett, Carbery, Christ, Tao and of Valdimarsson have clarified when equality holds in the Brascamp-Lieb inequality. Here we characterize the case of equality in the Geometric case of Barthe's reverse Brascamp-Lieb inequality.

Rényi entropy power inequality and a reverse

This paper is twofold. In the first part, we present a refinement of the Rényi Entropy Power Inequality (EPI) recently obtained in \cite{BM16}. The proof largely follows the approach in \cite{DCT91} of employing Young's convolution inequalities with sharp constants. In the second part, we study the reversibility of the Rényi EPI, and confirm a conjecture in \cite{BNT15, MMX16} in two cases. Connections with various $p$-th mean bodies in convex geometry are also explored.

A Complementary Inequality to the Information Monotonicity for Tsallis Relative Operator Entropy

We establish a reverse inequality for Tsallis relative operator entropy involving a positive linear map. In addition, we present converse of Ando's inequality, for each parameter. We give examples to compare our results with the known results by Furuta and Seo. In particular, we establish an extension and a reverse of the Löwner-Heinz inequality under certain condition. Some interesting consequences of inner product spaces and norm inequalities are also presented.

Reverses of the Young inequality for matrices and operators

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 …

A note one Sobolev type inequalities with p<1 and modified norm

A note that points out the possibility to have p<1 in Sobolev type of inequalities by a use of the momomial structure of polynomials or power series. The proof is simple: Triangle angle inequality p*>1, monomial estimate from p* to exponent p<1, then the inverse triangle inequality.

Reverses and variations of Heinz inequality

Let $A, B$ be positive definite $n\times n$ matrices. We present several reverse Heinz type inequalities, in particular \begin{align*} \|AX+XB\|_2^2+ 2(ν-1) \|AX-XB\|_2^2\leq \|A^νXB^{1-ν}+A^{1-ν}XB^ν\|_2^2, \end{align*} where $X$ is an arbitrary $n \times n$ matrix, $\|\cdot\|_2$ is Hilbert-Schmidt norm and $ν>1$. We also establish a Heinz type inequality involving the Hadamard product of the form \begin{align*} 2|||A^{1\over2}\circ B^{1\over2}|||\leq|||A^{s}\circ B^{1-t}+A^{1-s}\circ B^{t}||| \leq\max\{|||(A+B)\circ I|||,|||(A\circ B)+I|||\}, \end{align*} in which $s, t\in [0,1]$ and $|||\cd…

An extension of Milman's reverse Brunn-Minkowski inequality

The classical Brunn-Minkowski inequality states that for $A_1,A_2\subset\R^n$ compact, $$ |A_1+A_2|^{1/n}\ge |A_1|^{1/n}+|A_2|^{1/n}\eqno(1) $$ where $|\cdot|$ denotes the Lebesgue measure on $\R^n$. In 1986 V. Milman {\bf [Mil 1]} discovered that if $B_1$ and $B_2$ are balls there is always a relative position of $B_1$ and $B_2$ for which a perturbed inverse of $(1)$ holds. More precisely:\lq\lq{\sl There exists a constant $C>0$ such that for all $n\in\N$ and any balls $B_1,B_2\subset\R^n$ we can find a linear transformation $u\colon\R^n\to\R^n$ with $|{\rm det}(u)|=1$ and $$|u(B_1)+B_2|^{1/n…

Reverse Hölder Inequalities for log-Lipschitz Functions

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…

Reverse Holder, Minkowski, And Hanner Inequalities For Matrices

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…

关于独立同分布对数凹随机变量的反向熵幂不等式

A reverse entropy power inequality for i.i.d. log-concave random variables

研究独立同分布对数凹随机变量的反向熵幂不等式，通过递减重排、控制和测度变换等工具，证明\(h_\infty(X + Y) \leq h_\infty(Z + W)\)，并得到整数值单调对数凹随机变量类似结果。