$\log(f(z))$ analytic on closed disk

Question

I would like to have a function $f: \mathbb{C} \to \mathbb{D}$ such that $\log(f)$ is analytic on some disk $|z| \leq r$ for some $0<r<1$. Clearly $f(z)=k$ for some constant $k$ would work as $\log(f(z))$ then would be constant. However I am wondering what assumptions that need to be made on $f$ in orther to be sure that $\log(f)$ is analytic on $|z| \leq \rho$. I would also be very happy for some examples for function that satisfies this that is a bit more intresting than a constant function.

Answer 1

By Liouville's theorem there's no nonconstant holomorphic function from $\Bbb C$ to $\Bbb D$.