That is, is it true that the bound 
$$\phi(mn)\leq m\phi(n)$$ holds for all pairs of natural numbers $m$ and $n$?


It is true on average, in the sense that 
$$\sum\_{mn=k}m\phi(mn)\leq\sum\_{mn=k}mn\phi(n),$$ and it holds for every pair I have computed. If it does not hold, what is the smallest counter example? 



Is Euler's totient sub-homogeneous of degree 1?
First note that, since $\phi$ is a multiplicative arithmetic function, i.e. $\phi(ab)=\phi(a)\phi(b)$ for coprime $a,b$, and since $\phi(a) \le a$ for every $a$, it suffices to consider the problem for prime powers.


And $$\phi(p^k p^t) = p^{k+t-1}(p-1) = p^k (p^{t-1}(p-1))= p^k\phi(p^t).$$


