相关论文: A matrix subadditivity inequality for f(A+B) and f…

Mercer inequality for convex functions is a variant of Jensen's inequality, with an operator version that is still valid without operator convexity. This paper is two folded. First, we present a Mercer-type inequality for operators without…

A famous result of Lieb establishes that the map $(A,B) \mapsto \text{tr}\left[K^* A^{1-t} K B^t\right]$ is jointly concave in the pair $(A,B)$ of positive definite matrices, where $K$ is a fixed matrix and $t \in [0,1]$. In this paper we…

According to a celebrated result by L\"owner, a real-valued function $f$ is operator monotone if and only if its L\"owner matrix, which is the matrix of divided differences $L_f=(\frac{f(x_i)-f(x_j)}{x_i-x_j})_{i,j=1}^N$, is positive…

Suppose that $f$ belongs to a suitably defined complete metric space $ {{\cal C}}^{{\alpha}}$ of H\"older $ {\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper…

In this paper we consider a generalized version of Carleman's inequality. An equivalent version of it states that $\|f\|_{A_\alpha^{2\alpha}}\leq\|f\|_{H^2}$, where $f$ is a holomorphic function and $\alpha>1$. If the norms…

A Minkowski symmetral of an $\alpha$-concave function is introduced, and some of its fundamental properties are derived. It is shown that for a given $\alpha$-concave function, there exists a sequence of Minkowski symmetrizations that…

The Brunn-Minkowski Theorem asserts that $\mu_d(A+B)^{1/d}\geq \mu_d(A)^{1/d}+\mu_d(B)^{1/d}$ for convex bodies $A,\,B\subseteq \R^d$, where $\mu_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and…

We will show that if $\sM$ is a factor, then for any pair $\f, \p\in\sMdsup$ of normal positive linear functionals on $\sM$, the inequality: $$ \lrnorm{\f}\leq \lrnorm{\p} $$ is equivalent to the fact that there exist a countable family…

We obtain an interesting inequalities for uniformly continuous functions in the normed spaces: $\|f(x)\|\leq a\|x\|+b$ for some $a,b> 0$.

We show that the family of all holomorphic functions $f$ in a domain $D$ satisfying $$\frac{|f^{(k)}|}{1+|f|}(z)\le C \qquad \mbox{ for all } z\in D$$ (where $k$ is a natural number and $C>0$) is quasi-normal. Furthermore, we give a general…

Let $A,B\in B(H)$. We present among others a simple proof of the widely known result stating that if $0\leq A\leq B$, then $\sqrt A\leq \sqrt B$. The same idea is used to prove that if $0\leq A\leq B$ and $A$ is invertible, then $B$ too is…

The matrix convexity and the matrix monotony of a real $C^1$ function $f$ on $(0,\infty)$ are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with $f$, $tf(t)$, and $t^2f(t)$.…

We prove a functional extension of an exponential inequality originally proposed by Bin Zhao and proved by Xiaosheng Mou. The main result asserts that if $\alpha_1\leq \cdots\leq \alpha_n$ and $\sum_{k=1}^n \alpha_k=0$, then \[ \sum_{k=1}^n…

We prove that for a positive integer $c$ and any given $\varepsilon$, $0<\varepsilon<1$, the number $N(c)$ of equations $c=a+b$, $a<b$, with positive coprime integers $a$ and $b$, which satisfy the inequality $$c <…

A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the…

We consider functions $f(v)=\min_{A\in K}{Av}$ and $g(v)=\max_{A\in K}{Av}$, where $K$ is a finite set of nonnegative matrices and by "min" and "max" we mean coordinate-wise minimum and maximum. We transfer known results about properties of…

Given a pair $A,B$ of matrices of size $n\times n$, we consider the matrix function $e^{At+B}$ of the variable $t\in\mathbb{C}$. If the matrix $A$ is Hermitian, the matrix function $e^{At+B}$ is representable as the bilateral Laplace…

Suppose that A and B are real stable matrices, and that their difference A-B is rank one. Then A and B have a common quadratic Lyapunov function if and only if the product AB has no real negative eigenvalue. This result is due to Shorten…

Eigenvalues inequalities involving (log) convex/concav functions and Hermitian matrices, positive unital maps are considered. Simple proofs of Bhatia-Kittaneh inequality and Naimark dilation theorem are given.

In this work, we introduce a new class of non-convex functions, called implicit concave functions, which are compositions of a concave function with a continuously differentiable mapping. We analyze the properties of their minimization by…