I am working on this problem:
Let $U\subset \mathbb{C}$ be a domain is bounded by two disjoint simply closed Jordan curves $J_1$ and $J_2$. Let $u$ be a function that is continuous in $\overline{U}$, harmonic in $U$ and $$u(z)=\begin{cases}0 &z\in J_1, \\-1 &z\in J_2.\end{cases}$$ Show that there is a real number $c$ such that $$f(z)=\exp \left(c\int_{z_0}^{z} du+i\,\ast\, du\right )$$ maps $U$ biholomorphically onto an annulus. Here $z_0$ is a fixed point in $U$, and the path of integration is arbitrary within $U$.
All I know is $$\int_{z_0}^{z} du+i\,\ast\, du$$ is a holomorphic function (where $\ast$ is the Hodge operator). So $f$ is still holomorphic.
I have no idea about the injectivity and surjectivity.
Maybe we can use $c$ to make this map injective, that is, to make the imaginary parts of the numbers $$c\int_{z_0}^{z} du+i\,\ast\, du$$ less than $2\pi i$.