Jean-Christophe Bourin
Matrix versions of some basic convexity inequalities are given. Further results on the same topic are proved in the recent papers on arxiv: 1. Hermitian operators and convex functions, 2. A concavity inequality for symmetric norms, 3. Eigenvalue inequalities for convex and log-convex functions (with Jaspal Singh AUJLA).
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Convexity and concavity of $f$-potentials (Kolmogorov means)
本文给出$f$-势（Kolmogorov均值）凸性与凹性的判别准则，涵盖算术、几何、调和均值及$L^p$范数等特例，并用求积法求出所有满足条件的函数$f$。
Duality between Y-convexity and $Y^{\times}$-concavity of linear operators between Banach lattices
本文研究了线性算子的Y-凸性，通过引入向量序列空间并利用相关算子的连续性对其进行刻画，同时得出了Y-凹性的类似结果，并最终证明了Y-凸性与$Y^{\times}$-凹性之间的对偶性。
In this paper, we obtain some new estimations of Iyengar-type inequality in which quasi-convex(quasi-concave) functions are involved. These estimations are improvements of some recently obtained estimations. Some error estimations for the trapezoidal formula are given. Applications for special means are also provided.
A generalization of p-convexity and q-concavity on Banach lattices
该研究以实 Banach 序列格 Y 替代 $l_p$，将线性算子的 p-凸性推广为 Y-凸性，同时推广 q-凹性与 p-可和性，并验证相关基本性质及复合关系。
Convexity and concavity in Banach lattices
本文系统介绍Banach格中(p,q)-凸性、(p,q)-凹性及上下p-估计等概念，阐述p-凸化与p-凹化技术，并给出相关算子的主要因子分解结果及其在重赋范与表示中的应用。
We review some convexity inequalities for Hermitian matrices an add one more to the list.
We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of the extended Lieb type $Tr{Φ(A^p)^{1/2}Ψ(B^q)Φ(A^p)^{1/2}}^s$, where $Φ$ and $Ψ$ are positive linear maps. By the same method combined with majorization technique, similar properties are proved for symmetric (anti-) norm functions of the form $||{Φ(A^p)σΨ(B^q)}^s||$ involving an operator mean $σ$. Carlen and Lieb's variational method is also used to improve the convexity property of norm functions $||Φ(A^p)^s||$.
Employing the notion of operator log-convexity, we study joint concavity$/$ convexity of multivariable operator functions: $(A,B)\mapsto F(A,B)=h\left[ Φ(f(A))\ σ Ψ(g(B))\right]$, where $Φ$ and $Ψ$ are positive linear maps and $σ$ is an operator mean. As applications, we prove jointly concavity$/$convexity of matrix trace functions $\Tr\left\{ F(A,B)\right\}$. Moreover, considering positive multi-linear mappings in $F(A,B)$, our study of the joint concavity$/$ convexity of $(A_1,\cdots,A_k)\mapsto h\left[ Φ(f(A_1),\cdots,f(A_k))\right]$ provides some generalizations and complement to results o…
In this article the operator trace function $ Λ_{r,s}(A)[K, M] := {\operatorname{tr}}(K^*A^r M A^r K)^s$ is introduced and its convexity and concavity properties are investigated. This function has a direct connection to several well-studied operator trace functions that appear in quantum information theory, in particular when studying data processing inequalities of various relative entropies. In the paper the interplay between $Λ_ {r,s}$ and the well-known operator functions $Γ_{p,s}$ and $Ψ_{p,q,s}$ is used to study the stability of their convexity (concavity) properties. This interplay may…
Equality cases in monotonicity of quasi-entropies, Lieb's concavity and Ando's convexity
本文重新研究了Petz拟熵的单调性，并系统刻画了拟熵数据处理不等式、Lieb凹性及Ando凸性中等式成立的等价条件，同时将结果推广到单调度量与χ²-散度，并讨论了相关量子信息量的线性保持问题。
We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of Lieb type $\mathrm{Tr}\,f(Φ(A^p)^{1/2}Ψ(B^q)Φ(A^p)^{1/2})$ and symmetric (anti-) norm functions of the form $\|f(Φ(A^p)\,σ\,Ψ(B^q))\|$, where $Φ$ and $Ψ$ are positive linear maps, $σ$ is an operator mean, and $f(x^γ)$ with a certain power $γ$ is an operator monotone function on $(0,\infty)$. Moreover, the variational method of Carlen, Frank and Lieb is extended to general non-decreasing convex/concave functions on $(0,\infty)$ so that we prove joint concavity/convexity of more trace functions of Lieb typ…
Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this p…
In this paper we will develop a systematic method to answer the questions $(Q1)(Q2)(Q3)(Q4)$ (stated in Section 1) with complete generality. As a result, we can solve the difficulties $(D1)(D2)$ (discussed in Section 1) without uncertainty. For these purposes we will introduce certain classes of growth functions $u$ and apply the Legendre transform to obtain a sequence which leads to the weight sequence $\{\a(n)\}$ first studied by Cochran et al. \cite{cks}. The notion of (nearly) equivalent functions, (nearly) equivalent sequences and dual Legendre functions will be defined in a very natural …
We obtain operator concavity (convexity) of some functions of two or three variables by using perspectives of regular operator mappings of one or several variables. As an application, we obtain, for $ 0<p < 1,$ concavity, respectively convexity, of the Frechét differential mapping associated with the functions $t \to t^{1+p} $ and $t \to t^{1-p}.$
We survey some interplays between spectral estimates of Hörmander-type, degenerate Monge-Ampère equations and geometric inequalities related to log-concavity such as Brunn-Minkowski, Santaló or Busemann inequalities.
We prove Lieb type convexity and concavity results for trace functionals associated with positive operator monotone (decreasing) functions and certain monotone concave functions. This gives a partial generalization of Hiai's recent work on trace functionals associated with power functions.
We establish the log-concavity of the volume of central sections of dilations of the cross-polytope (the strong B-inequality for the cross-polytope and Lebesgue measure restricted to an arbitrary subspace).
It is known that the volume functional $\,ϕ\mapsto\int e^{-ϕ}\,$ satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on ${\rm{Cvx}}_0({\mathbb R}^n)$. In this note we define the fourth linear structure on ${\rm{Cvx}}_0({\mathbb R}^n)$ as the pullback of the standard linear structure under the ${\cal J}$ transform. We show that, interpolating with respect to this linear structure, no concavity or convexity inequalities hold, and prove that a quasi-convexity inequality is violated only by up to a facto…
Several inequalities for eigenvalues involving convex combinations and compressions are given. These inequalities are matrix version of the basic convexity inequality f((a+b)/2) < (f(a)+f(b))/2.
Given a positive function $f$ on $(0,\infty)$ and a non-zero real parameter $θ$, we consider a function $I_f^θ(A,B,X)=Tr X^*(f(L_AR_B^{-1})R_B)^θ(X)$ in three matrices $A,B>0$ and $X$. In the literature $θ=\pm1$ has been typical. The concept unifies various quantum information quantities such as quasi-entropy, monotone metrics, etc. We characterize joint convexity/concavity and monotonicity properties of the function $I_f^θ$, thus unifying some known results for various quantum quantities.
The main purpose of this paper is the study of a~new class of summing multilinear operators acting from the product of Banach lattices with some nontrivial lattice convexity. A~mixed Pietsch-Maurey-Rosenthal type factorization theorem for these operators is proved under weaker convexity requirements than the ones that are needed in the Maurey-Rosenthal factorization through products of $L^q$-spaces. A~by-product of our factorization is an extension of multilinear operators defined by a~$q$-concavity type property to a~product of special Banach function lattices which inherit some lattice-geome…
In this paper, we introduce two notions of a relative operator $(α, β)$-entropy and a Tsallis relative operator $(α, β)$-entropy as two parameter extensions of the relative operator entropy and the Tsallis relative operator entropy. We apply a perspective approach to prove the joint convexity or concavity of these new notions, under certain conditions concerning $α$ and $β$. Indeed, we give the parametric extensions, but in such a manner that they remain jointly convex or jointly concave.
Let $T^m_f $ be the Toeplitz quantization of a real $ C^{\infty}$ function defined on the sphere $ \mathbb{CP}(1)$. $T^m_f $ is therefore a Hermitian matrix with spectrum $λ^m= (λ_0^m,\ldots,λ_m^m)$. Schur's theorem says that the diagonal of a Hermitian matrix $A$ that has the same spectrum of $ T^m_f $ lies inside a finite dimensional convex set whose extreme points are $\{( λ_{σ(0)}^m,\ldots,λ_{σ(m)}^m)\}$, where $σ$ is any permutation of $(m+1)$ elements. In this paper, we prove that these convex sets "converge" to a huge convex set in $L^2([0,1])$ whose extreme points are $ f^*\circ ϕ$, wh…
In this paper we study the joint convexity/concavity of the trace functions \[ Ψ_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, \] where $A$ and $B$ are positive definite matrices and $K$ is any fixed invertible matrix. We will give full range of $(p,q,s)\in\mathbb{R}^3$ for $Ψ_{p,q,s}$ to be jointly convex/concave for all $K$. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of $(α,z)$ for $α$-$z$ Rényi relative entropies to be monotone u…
Some convexity and monotonicity results of trace functionals
本文证明了形如 Tr|B^p A C^q|^s 的迹泛函在最优参数范围内的凸性，并得到其单调性版本，进而推广已有结果、解决矩阵情形下的一个猜想，同时指出某些相关迹泛函一般不具有凹性。
Linear matrix pencils and noncommutative convexity
本综述从自由实代数几何视角阐述自由谱体的凸性本质，介绍其检测与线性矩阵束表示方法，并说明其正置定理可用于构建非交换约束下凸矩阵集特征值优化的半定规划形式。
Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities
本文研究Fisher-Rao度量下的梯度流动力学，探讨测地凸性与函数不等式，提出不依赖目标分布log-concavity或log-Sobolev常数的分析方法，确保动力学收敛速率在一般分布下保持一致。
We show that for convex domains in Euclidean space, Cheeger's isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have \emph{arbitrarily slow} uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz'ya, Cheeger, Gromov--Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under con…
From Majorization to Horn and Unitary-Orbit Inequalities
本文提出一个统一框架，将优超不等式提升为完整的Horn不等式（等价于酉轨道不等式），并应用于三类算子不等式，得到推广的酉轨道不等式。
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