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Let $n \in \N$ and $M_n$ be the algebra of $n \times n$ matrices. We call afunction $f$ matrix monotone of order $n$ or $n$-monotone in short whenever theinequality $f(a) \leq f(b)$ 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 containedin $I$. Matrix convex (concave) functions on $I$ are similarily defined. Thespaces 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 thosesequences $\{P_n(I)\}_{n\in\N}$ and $\{K_n(I)\}_{n\in\N}$. In order to seeclear insight of the aspect of the problems, however, we choose the followingthree main assertions among them and discuss their mutual dependence:  \begin{enumerate} \item[(i)] $f(0)\leq 0$ 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 $M_n$,  \[ f(c^{\star}a c)\leq c^{\star}f(a)c, \] \item[(iii)] The functon $g(t)/t$is $n$-monotone in $(0,\alpha)$. \end{enumerate}  In particular, we show that for any $n \in \N$ two conditions $(ii)$ and$(iii)$ are equivalent.

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