If $\varrho$ is a metric and $\phi$ is concave then is $\phi \circ \varrho$ also a metric? The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated.
Let $\varrho: X\times X\to \Bbb R^+$ be a metric on $X$ and let $\phi: \Bbb R^+ \to \Bbb R^+$ be a function such that


*

*$\phi(0)=0$

*$\phi$ is strictly increasing, i.e. for $x_1<x_2$ we have $\phi(x_1)<\phi(x_2)$

*$\phi$ is concave, i.e. for $0\le\alpha\le 1$ and $x_1<x_2$
$$
\phi(\alpha x_1+(1-\alpha)x_2)\ge\alpha \phi(x_1)+(1-\alpha)\phi(x_2).
$$
Then $\phi\circ\varrho$ is a metric. [Hint: show that $\phi$ is sub-additive, that is 
$$
\phi(a+b)\le\phi(a)+\phi(b)
$$
for $a,b\ge0$].
 A: Of all the properties of a distance, the one requiring some work is the triangle inequality. Before tackling it, note the following: in (3) take $\alpha = \frac 1 2, x_2=0, x_1=x$. Then $\phi (\frac 1 2 x) \geq \frac 1 2 \phi (x)$ (using also (1)).
Since $\phi$ is increasing and $\rho$ satisfies the triangle inequality , $\phi \circ \rho (x,y) = \phi (\rho (x,y)) \leq  \phi (\rho(x,z) + \rho(z,y)) \leq 2 \phi (\frac 1 2 \rho(x,z) + \frac 1 2 \rho(z,y)) \leq 2(\frac 1 2 \phi (\rho(x,z)) + \frac 1 2 \phi(\rho(z,y))) = \phi (\rho (x,z)) + \phi (\rho(z,y)) = \phi \circ \rho (x,z) + \phi \circ \rho (z,y).$
Note that in the chain above, the 2nd "$\leq$" sign comes from what we have shown in the 1st paragraph.
EDIT: As shown in the comments below, this "proof" contains a mistake (it has been used that $\phi$ is convex, when in fact it is concave). For the correct proof see @Zac's comment below.
A: We have 3 way to prove it,way one:definition of derivation .way two: is 
mean theorem for sub interval,way three is definition of concave function (this way is hard).we prove from way two :because $\varphi$ is concave then $\varphi^\prime$ is descending ,and apply mean theorem for intervals [0,a] and [b,b+a] assume $0<a<b$ ,for example the function $\sqrt{x}$ is strictly increasing and concave ,(you draw this function for help and apply mean theorem).
