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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 label $n$ are discussed: (1)$f(0) \\leq 0$ and $f$ is $n$-convex in $[0, \\alpha)$. (2)For each matrix $a$ with its spectrum in $[0, \\alpha)$ and a contraction $c$ in the matrix algebra $M_n$, $f(c^*ac) \\leq c^*f(a)c$. (3)The function $f(t)/t$ $(= g(t))$ is $n$-monotone in $(0, \\alpha)$. 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