Let $F(z)$ be an inner function in the upper half plane, (i.e. $F$ is bounded analytic function such that $\lim_{y\to 0^{+}}|F(x+iy)|=1$ for almost all $x\in \mathbb R$ with respect to the Lebesgue measure). I need to prove that:
If $F$ admits an analytic extension across the real axis (hence is meromorphic in the whole complex plane) then there is a well-defined branch of the argument of $F$ on the line, i.e., an increasing differentiable function $\psi$ such that $F(x)=\exp(i \psi(x)), x\in \mathbb R$.