# Upper bound for series related to an injective holomorphic function on the unit disc

The following is part of an exercise that I don't quite get:

Let $\phi: \mathbb D\setminus \{0\} \to \mathbb C$ be an injecitve holomorphic function with Laurent series $$\phi(z) = \frac1z + \sum_{n=1}^\infty a_nz^n$$ Prove that $$\sum_{n=1}^\infty n|a_n| \le 1$$

Hint: Calculate the area of $S_r = \phi(B_1(0)\setminus B_r(0))$ using Stokes' thoerem and consider the limit as $r \to 1$.

I have to say, I'm a bit clueless here. I don't even really see how to calculate the area of $S_r$. It seems that Stokes' theorem forces me to consider the real and imaginary parts of $\phi$ and their partial derivatives, but it seems a bit messy to write these in terms of $\phi$ etc. What I've tried:

Writing $\phi = u + iv$ we get

$$\begin{eqnarray} \iint_{S_r} dx\, dy &=& \int_{\partial\phi(B_1\setminus B_r)}x\ dy \\ &=& \int_{\partial B_1- \partial B_r} uv_x \ dx + uv_y \ dy \end{eqnarray}$$

If I go on from here by substituting $u = \frac12(\phi + \overline \phi), v = \frac12(\phi - \overline \phi)$, then the result I come up with for the integral over $\partial B_r(0)$ seems to be

$$\pi\left(\frac{-1}{r^2} + \sum_{n=1}^\infty n|a_n|^2 r^{2n}\right)$$

But I'm neither really sure whether this is right (I'm guessing not, because the series has the wrong form), nor do I know what to do with that.

So my question really is:

• What is the best approach to this problem? Any hints/solutions would be appreciated.
• What is the motivation behind considering $S_r$ (probably a correct result would answer this retrospectively)?

• I suspect an error in the problem, because there is a very similar proof in Carleson and Gamelin's "Complex dynamics", p.2. The theorem they state is with the sum of the squares of modulus of $a_n$ just like what you obtain Dec 6, 2011 at 19:38