Integral of derivative of rational map on unit disk

Question

Let $f:D \rightarrow D$ be a surjective rational map of the unit disk of degree $n$. Prove that $$\iint_D |f'(x+iy)|\:\mathrm{d}x\:\mathrm{d}y\leq \pi \sqrt{n}.$$

Attempt:

We know that rational maps of degree $n$ can be thought of as covering maps since every point except "critical points" has $n$ pre-images. Also know that $|f'|^2$ gives local area change. Not sure how to tie this together.

Answer 1

The Cauchy-Schwarz inequality gives you

$$\iint_D \lvert f'(x+iy)\rvert\,dx\,dy \leqslant \biggl(\iint_D 1^2\,dx\,dy\biggr)^{1/2}\cdot\biggl(\iint_D \lvert f'(x+iy)\rvert^2\,dx\,dy\biggr)^{1/2}.$$

What remains is to see that

$$\iint_D \lvert f'(x+iy)\rvert^2\,dx\,dy = n\pi.$$

But that follows since $\lvert f'\rvert^2$ is the Jacobian determinant of $f$, and $f$ is an $n$-sheeted branched covering of $D$.