Representation of Holomorphic Functions By Exponential
Solution 1:
Yes, this is true on any simply connected domain. In this case you can take $$h(z) = h(0) + \int_C \dfrac{f'(\zeta)}{f(\zeta)}\ d\zeta = h(0) + \int_0^1 \dfrac{f'(tz)}{f(tz)} z\ dt $$ where $C$ is the straight line from $0$ to $z$ and $h(0)$ is any branch of $\log(f(0))$.