In An Introduction to Harmonic Analysis by Katznelson I once stumbled upon this:
with the footnote
Here $D$ is the complex unit disk around $0$. $f$ is holomorphic with no zeros, and since $D$ is simply connected, we have that some branch $\log(f)$ is holomorphic as proved here. So now we can define $g(z)=(f(z))^\frac{p}{2}:=\exp(\frac{p}{2}\log(f(z)))$.
Now Katznelson claims that we can take any branch. I used to ignore this part, but the more I think about it, the more I get suspicious. What if the range of $f$ has a point in the branch cut? Then this should not work.
Am I wrong, or does he mean some branch?
If $f$ has no zero in the unit disk, then ANY branch of $\log f$ and of $f^\alpha$ is analytic. This follows from Monodromy Theorem (look in any course of Complex analysis). You actually cite this theorem when you write "simply connected".
(Branches of log differ only by an additive constant, if one is holomorphic then all are holomorphic). There is no such thing as a "branch cut":-) It is just an artificial tool, not something associated with a function. If you look at the proof of the monodromy theorem you see that there is no "branch cut" involved.