Biholomorphic maps from unit disc

Question

Let $f$ be biholomorphic map from the unit disc onto some $D \subset \overline{\mathbb{C}}$ (considered as a Riemann sphere, so it is holomorphic) with $$f(z)=\frac{1}{z}+c_1z+c_2z^2+\cdots$$ What does inequality $$\sum n |c_n|^2 \leq 1$$ mean geometrically?

Answer 1

The complement of $D$ is a compact subset of the plane. Its area can be computed in terms of $f$, using Green's formula. This computation yields $$\text{area of $\mathbb C\setminus D$} = \pi-\pi\sum_{n=1}^\infty n|c_n|^2$$ Therefore, the inequality $\sum_{n=1}^\infty n|c_n|^2 \le 1$ expresses the fact that area cannot be negative. This is why the result is known as the Area Theorem; the wiki has the detailed computation to which I alluded above.