Related papers: Double piling structure of matrix monotone functio…

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…

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 $t^2f(t)$.…

Given an nxn doubly stochastic matrix P satisfying an appropriate condition of linear algebraic-type, and a function f defined on a nonempty interval, we show that the validity of a convexity-type functional inequality for f in terms P…

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 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…

This paper concerns matrix "convex" functions of (free) noncommuting variables, $x = (x_1, \ldots, x_g)$. Helton and McCullough showed that a polynomial in $x$ which is matrix convex is of degree two or less. We prove a more general result:…

In a recent article (2022) we proved with L. Zaj\'i\v{c}ek that if $ G\subset\R^n $ is an unbounded open convex set that does not contain a translation of a convex cone with non-empty interior, then there exist $ f:G\to\R $ and a concave…

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…

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…

We prove a condition on f \in C^2(\R+,\R) for the convexity of (f o det) on PSym(n), namely that f o det is convex on PSym(n) if and only if f"(s)+(n-1)/(ns) f'(s) >= 0 and f'(s)<= 0 \forall s \in \R+. This generalizes the observation that…

In this paper we give an integral representation of an $n$-convex function $f$ in general case without additional assumptions on function $f$. We prove that any $n$-convex function can be represented as a sum of two $(n+1)$-times monotone…

In this paper we show that for a non-negative operator monotone function $f$ on $[0, \infty)$ such that $f(0)= 0$ and for any positive semidefinite matrices $A$ and $B$, $$ Tr((A-B)(f(A)-f(B))) \le Tr(|A-B|f(|A-B|)). $$ When the function…

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal…

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…

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…

Let $\mathcal A$ be an $\mathbb F$-algebra and $\omega \in \mathcal A\langle x_1, \ldots, x_m \rangle$ which defines a map $\mathcal A^m \rightarrow \mathcal A$ by evaluation, called a polynomial map with constant. We consider $\mathcal {A}…

We investigate polyconvexity of the double well function $f(X)\,:= |X-X\_1|^2|X-X\_2|^2$ for given matrices $X\_1, X\_2 \in \R^{n \times n}$. Such functions are fundamental in the modeling of phase transitions in materials, but their…

Matrix extension of a scalar function of a single variable is well-studied in literature. Of particular interest is the trace of such functions. It is known that for diagonalizable matrices, $M$, the function $g(M) = \text{Tr}(f(M)) =…

One matrix structure in the area of monotone Boolean functions is defined here. Some of its combinatorial, algebraic and algorithmic properties are derived. On the base of these properties, three algorithms are built. First of them…

Let f be a non-negative concave function on the positive half-line. Let A and B be two positive matrices. Then, for all symmetric norms, || f(A+B) || is less than || f(A)+f(B) ||. When f is operator concave, this was proved by Ando and…