I trying to understand a proof in Pommerenke's Univalent Functions. Given $f$ analytic on the unit disk and $M(r) := \max\limits_{0 \leq \theta \leq 2\pi}|f(re^{i\theta})|$,
Using the fact that $M(r) \leq \frac{r}{(1-r)^2},$ $(0 < r < 1)$, I want to show that $\int_0^1 \frac{M^p(r)}{r}dr < \infty,$ for $0 < p < \frac{1}{2}$.
I get $\frac{M^p(r)}{r} ~\leq~ \frac{r^{p-1}}{(1-r)^{2p}}.$
The term on the RHS doesn't seem easy to integrate on $(0,1)$...