In Ahlfors' text, he considers a region $\Omega$ bounded by analytic contours $C_1, \dots, C_n$.
On page 259 he states:
We have thus established the existence of a function $p(z)$ which is single-valued and analytic in $\Omega$, except for a simple pole with the residue $1$ at $z_0$, and whose real part is constant on each contour. By these requirements $p(z)$ is uniquely determined up to an additive constant.
I can't see how do the requirements determine $p(z)$: If $p_1(z)$ is another function with these properties, we have that $p-p_1$ has no singular part, and still has its real part constant on each contour. I can't see why this difference must reduce to a constant though.
Could you please help me see why it is the case?
Thanks!
Besides other things, the function $p$ is continuous on $\overline{\Omega}$. If $p_1$ is another one such, and $f:=p-p_1$ is nonconstant, then $f(\Omega)$ is a bounded domain. By the open mapping theorem $\partial f(\Omega)\subset f(\partial \Omega)$. But the values that $f$ takes on $\partial \Omega$ lie on a finite union of vertical lines. Being open, $ f(\Omega)$ contains a point $w$ which is not on any of those lines. The vertical line through $w$ does not meet $\partial f(\Omega)$, which contradicts the latter being bounded.