Let $f(x)$ be a real valued function of real variable, define for all $x \geq 0$. $f$ is said to be concave if for any nonnegative real numbers a,b with $a+b=1$ we have
$$f(ax+by)\geq af(x)+bf(y)$$
Suppose that $f(x)$ is concave, that $f(0)=0$, that $f(x)>0$ for $x>0$, and that $f$ is monotone in the weak sense ($x\leq y \Rightarrow f(x) \leq f(y)$). Let $M$ be a metric space with metric $d$. Prove that $f(d)$ is also a metric of $M$.
I'd like some hints on how to prove the triangular inequality for $f(d)$, I've tried to apply $f$ in the triangular inequality for $d$ getting $f(d(a,b))\leq f(d(a,c)+d(b,c)$, but I don't know how to proceed.
