I am studying matrix monotone and I meet some difficulties when trying to prove the following statement:
Let $f$ be a differentiable function on the interval $(a,b)$ such that $f$ is 2-monotone on $(a,c]$ and $[c,b)$ for some $a < c < b$. Prove that $f$ is 2-monotone on $(a,b)$.
(Note: A function $f$ is said to be $n$-monotone if $f$ is matrix monotone for $n\times n$ matrices.)
I know from the textbook of Hiai and Petz that if $f$ is 2-monotone on $(\alpha,\beta)$ then the matrix $D$ defined as $$ D = \left[\begin{array}{cc}f^{\prime}(x) & \frac{f(x)-f(y)}{x-y} \\ \frac{f(x)-f(y)}{x-y} & f^{\prime}(y)\end{array}\right] $$ is positive semidefinite for $x, y \in (\alpha,\beta)$, $x\neq y$.
I want to show that if $D\geq 0$ for $x, y \in (a,c]$ and $[c,b)$, respectively, then $D\geq 0$ for $x, y \in (a,b)$. However, I cannot prove this.
Any suggestions? Or is there an alternative way to prove the original statement?