Let $U$ denote the open annulus $\{z\in \Bbb C:1<|z|<2\}$ and suppose $f:U\to U$ is a holomorphic bijection. I want to show that $u(z)=2\log |f(z)|-2\log |z|$ is a harmonic function on $U$.
Clearly harmonic is a local property, so it suffices to show that $u$ is harmonic at each point of $U$. Also a function is harmonic in an open set iff it is locally the real part of a holomorphic function. Thus I tried to show that $u$ is locally the real part of a holomorphic function. The definition of $u$ suggests to consider the function $f(z)/z$ (because $u(z)=2\log |f(z)/z|$, which is nonzero and holomorphic in $U$, and satisfies $1/2 \leq |f(z)/z|\leq 2$. But I have no idea then. Any hints?
HINT:
Let $f(z)=u(x,y)+iv(x,y)$ where $u$ and $v$ satisfy the Cauchy-Riemann equations and are, therefore harmonic.
Now just show that
$$\left( \frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right)\left( \frac{\partial }{\partial x}-i\frac{\partial }{\partial y}\right) \log(u^2(x,y)+v^2(x,y))=0$$