Structure of the spaces of matrix monotone functions and of matrix convex functions and Jensen's type inequality for operators

Let $n\in \mathbb{N}$ 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 $f(a)\le f(b)$ holds for every pair of selfadjoint matrices $a, b \in M_n$suchthat$a \leq b$andalleigenvaluesof$a$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 piling structure of those sequences $\{P_n(I){\}}_{n\in \mathbb{N}}$ and $\{K_n(I){\}}_{n\in \mathbb{N}}$. In order to see clear insight of the aspect of the problems, however, we choose the following three main assertions among them and discuss their mutual dependence: \begin{enumerate} \item[(i)] $f(0)\le 0$ and $f$ is $n$-convex in $[0,\alpha )$, \item[(ii)] For each matrix $a$ with its spectrum in $[0,\alpha )$ and a contraction $c$ in the matrix algebra $M_n$, $$f(c^{\star }ac)\le c^{\star }f(a)c,$$ \item[(iii)] The functon $g(t)\mathrm{/}t$ is $n$-monotone in $(0,\alpha )$. \end{enumerate} In particular, we show that for any $n\in \mathbb{N}$ two conditions $(ii)$ and $(iii)$ are equivalent.