Solution:
We have to prove that
$$\frac{u^{q}-v^{q}}{1-(u v)^{q}} \leq \frac{u^{p}-v^{p}}{1-(u v)^{p}}$$
for $v \leq u$ and $p<q$. To do this we shall show that
$$\frac{u^{p+1}-v^{p+1}}{1-(u v)^{p+1}} \leq \frac{u^{p}-v^{p}}{1-(u v)^{p}}$$
which implies by induction the above inequality. It is easy to check that the latter inequality can be written as
$$\frac{1-u^{2 p+1}}{u^{p}(1-u)} \leq \frac{1-v^{2 p+1}}{v^{p}(1-v)}$$
i.e.
$$\sum_{j=1}^{p}\left(u^{j}+u^{-j}\right) \leq \sum_{j=1}^{p}\left(v^{j}+v^{-j}\right)$$
which follows by the inequality $t+\frac{1}{t} \leq z+\frac{1}{z}$, where $0<z \leq t \leq 1$.