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 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 $(0,
\alpha)$. We know that two conditions $(2)$ and $(3)$ are equivalent and if $f$
with $f(0) \leq 0$ is $n$-convex, then $g$ is $(n -1)$-monotone. In this note
we consider several extra conditions on $g$ to conclude that the implication
from $(3)$ to $(1)$ is true. In particular, we study a class $Q_n([0, \alpha))$
of functions with conditional positive Lowner matrix which contains the class
of matrix $n$-monotone functions and show that if $f \in Q_{n+1}([0, \alpha))$
with $f(0) = 0$ and $g$ is $n$-monotone, then $f$ is $n$-convex. We also
discuss about the local property of $n$-convexity.Comment: 13page