Related papers: Loewner matrices of matrix convex and monotone fun…

We characterize real functions $f$ on an interval $(-\alpha,\alpha)$ for which the entrywise matrix function $[a_{ij}] \mapsto [f(a_{ij})]$ is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional…

We give new characterizations for matrix monotonicity and convexity of fixed order which connects previous characterizations by Loewner, Dobsch, Donoghue, Kraus and Bendat--Sherman. The ideas introduced are then used to characterize matrix…

We establish local characterizations of matrix monotonicity and convexity of fixed order by giving integral representations connecting the Loewner and Kraus matrices, previously known to characterize these properties, to respective Hankel…

We generalize Loewner's method for proving that matrix monotone functions are operator monotone. The relation x \leq y on bounded operators is our model for a definition for C*-relations of being residually finite dimensional. Our main…

We study various convex functions on $R^n$ associated with positive definite matrices. This yiels some exotic Holder matrix inequalities.

We continue the analysis in [H. Osaka and J. Tomiyama, Double piling structure of matrix monotone functions and of matrix convex functions, Linear and its Applications 431(2009), 1825 - 1832] in which the followings three assertions at each…

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…

Operator monotone functions, introduced by Lowner in 1934, are an important class of real-valued functions. They arise naturally in matrix and operator theory and have various applications in other branches of mathematics and related…

Associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,\infty]^2\rightarrow[0,\infty]$ is an associative function, $f: [0,1]\rightarrow [0,\infty]$ is a monotone function…

We analyze matrix convex functions of a fixed order defined on a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus. We obtain for each order conditions for matrix…

Some real functions f induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define linear mapping beta on matrices (which is basic in the…

For an operator monotone function $f(t)$ on the positive real line, we show the operator monotonicity of the type of the functions $(t-a)(t-b)/(f(t)-f(a))(f^\sharp(t)-f^\sharp(b))$.

Let $ f:(0,\infty)\rightarrow \Bbb{R} $ be a completely monotonic function. In this paper, we present some properties of this functions and several new classes of completely monotonic functions. We also give some special functions such that…

In this paper, we prove the convexity of trace functionals $$(A,B,C)\mapsto \text{Tr}|B^{p}AC^{q}|^{s},$$ for parameters $(p,q,s)$ that are best possible, where $B$ and $C$ are any $n$-by-$n$ positive definite matrices, and $A$ is any…

This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and…

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…

We prove the existence of gaps between all the different classes of matrix monotone functions defined on an interval, provided the interval is non trivial and different from the whole real line. We then show how matrix monotone functions…

Given a positive function $f$ on $(0,\infty)$ and a non-zero real parameter $\theta$, we consider a function $I_f^\theta(A,B,X)=Tr X^*(f(L_AR_B^{-1})R_B)^\theta(X)$ in three matrices $A,B>0$ and $X$. In the literature $\theta=\pm1$ has been…

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of Lieb type $\mathrm{Tr}\,f(\Phi(A^p)^{1/2}\Psi(B^q)\Phi(A^p)^{1/2})$ and symmetric (anti-) norm functions of the form…

We continue the analysis in [3] of matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus [5]. We amend and improve some…