Frank Hansen, Josip Pecaric, Ivan Peric
We give a general formulation of Jensen's operator inequality for unital fields of positive linear mappings, and we consider different types of converse inequalities.
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Jensen's trace inequality is established for every multivariable, convex function and every trace or trace-like functional on a C*-algebra.
We study conditional expectations generated by an abelian $ C^* $-subalgebra in the centralizer of a positive functional. We formulate and prove Jensen's inequality for functions of several variables with respect to this type of conditional expectations, and we obtain as a corollary Jensen's inequality for expectation values.
We establish what we consider to be the definitive versions of Jensen's operator inequality and Jensen's trace inequality for functions defined on an interval. This is accomplished by the introduction of genuine non-commutative convex combinations of operators, as opposed to the contractions used in earlier versions of the theory. As a consequence, we no longer need to impose conditions on the interval of definition. We show how this relates to the pinching inequality of Davis, and how Jensen's trace inequlity generalizes to C*-algebras..
Classically, Jensen's Inequality asserts that if $X$ is a compact convex set, and $f:K\to \mathbb{R}$ is a convex function, then for any probability measure $μ$ on $K$, that $f(\text{bar}(μ))\le \int f\;dμ$, where $\text{bar}(μ)$ is the barycenter of $μ$. Recently, Davidson and Kennedy proved a noncommutative ("nc") version of Jensen's inequality that applies to nc convex functions, which take matrix values, with probability measures replaced by ucp maps. In the classical case, if $f$ is only a separately convex function, then $f$ still satisfies the Jensen inequality for any probability measu…
Let $\mathcal{A}$ be a $C^*$-algebra and $ϕ:\cA\to L(H)$ be a positive unital map. Then, for a convex function $f:I\to \mathbb{R}$ defined on some open interval and a self-adjoint element $a\in \mathcal{A}$ whose spectrum lies in $I$, we obtain a Jensen's-type inequality $f(ϕ(a)) \leq ϕ(f(a))$ where $\le$ denotes an operator preorder (usual order, spectral preorder, majorization) and depends on the class of convex functions considered i.e., operator convex, monotone convex and arbitrary convex functions. Some extensions of Jensen's-type inequalities to the multi-variable case are considered.
Jensen' s trace inequality with equality condition and noncommutative Lamperti's theorem
该研究针对半有限von Neumann代数，建立带等号条件的Jensen迹不等式，解答相关问题，推导非交换Lamperti型不等式，刻画一类F-范数非交换Orlicz空间上的线性等距，得到对应非交换Lamperti定理。
Jensen's inequality for partial traces in von Neumann algebras
本文在半有限冯·诺依曼代数中证明了部分追踪的詹森不等式，并扩展至一般冯·诺依曼代数框架。
Generalized Choi-Davis-Jensen's Operator Inequalities and Their Applications
针对原始Choi-Davis-Jensen不等式的推广需求，研究通过引入非线性映射、将算子凸函数拓展为紧区域上任意连续函数，借助Stone-Weierstrass定理与Kantorovich函数构建证明了广义不等式，并给出其在统计物理中的应用。
Let $n \in \N$ and $M_n$ be the algebra of $n \times n$ matrices. We call a function $f$ matrix monotone of order $n$ or $n$-monotone in short whenever the inequality $f(a) \leq f(b)$ holds for every pair of selfadjoint matrices $a, b \in M_n$ such that $a \leq b$ and all eigenvalues of $a$ and $b$ are contained in $I$. Matrix convex (concave) functions on $I$ are similarily defined. The spaces for $n$-monotone functions and $n$-convex functions are written as $P_n(I)$ and $K_n(I)$. In this note we discuss several assertions at each leven $n$ for which we regard themas the problems of double p…
Generalized Converses of Operator Jensens Inequalities with Applications to Hypercomplex Function Approximations and Bounds Algebra
本研究推广了Mond-Pecaric（MP）算子不等式方法，放宽凸函数与线性映射限制，推导超复函数新不等式，应用于函数逼近与界代数构建。
Jensen's inequality in finite subdiagonal algebras
本文将有限次对角代数中的Jensen不等式推广至更一般的递增连续函数情形，证明了迹不等式及其取等条件，并给出了关于点谱包含关系的应用。
We give an extension of Hua's inequality in pre-Hilbert $C^*$-modules without using convexity or the classical Hua's inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert $C^*$-modules, another extension of Hua's inequality is obtained. We also present an operator Hua's inequality, which is equivalent to operator convexity of given continuous real function.
A generalization of classical determinant inequalities like Hadamard's inequality and Fischer's inequality is studied. For a version of the inequalities originally proved by Arveson for positive operators in von Neumann algebras with a tracial state, we give a different proof. We also improve and generalize to the setting of finite von Neumann algebras, some `Fischer-type' inequalities by Matic for determinants of perturbed positive-definite matrices. In the process, a conceptual framework is established for viewing these inequalities as manifestations of Jensen's inequality in conjunction wit…
In this paper we introduce the notion of weak 2-positivity and present some examples. We establish some operator Cauchy--Schwarz inequalities involving the geometric mean and give some applications. In particular, we present some operator versions of Hua's inequality by using the Choi--Davis--Jensen inequality.
Improved Upper Bounds for Numerical Radius Inequalities of Operator Matrices and Their Applications
本研究运用多种不等式及Buzano不等式扩展形式，推导希尔伯特空间上算子矩阵数值半径的改进上界，完善经典结果，应用于量子力学等多领域的数学物理分析。
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