Is there a proof for area theorem, which does not use area argument?

Question

Area Theorem

Let $f(z)=z+b_0 + \frac{b_1}{z} + \frac{b_2}{z^2} + ... $ be an injective holomorphic function defined in the domain $|z|>1$.

Then, $\sum_{n=1}^\infty n|b_n|^2 \leq 1 $.

This theorem is known as the area theorem since its well-known proof uses Green's theorem.

I'm curious whether there is a more elementary way to prove this not using Green's theorem. Thank you in advance.