Let $F_0(z)=1/(1-z)^i$, verify that $|F_0(z)|\leq e^{\pi/2}$ in the unit disc, but that $\lim_{r\to 1} F_0(r)$ does not exist.
It's clear that $|F_0(r)|=1$, but I cannot recall how to do function with complex power. As I let $z=a+bi$ and use some equation like $x=\exp(\ln x)$, it becomes even more complicated.
Any help will be appreciated.
What do you know about logarithms in the complex plane? First thing you should know is that powers of complex numbers are not well-defined in general. Here, I believe $(1-z)^{i}$ is defined using the principal branch of logarithm. Write $F(z)=e^{-i Log (1-z)}$ and recall definition of $Log (z)$ as $\ln |z|+i\theta$ where $z=|z|e^{i \theta},-\pi < \theta < \pi$. Now can you complete the solution?