TITLE: Find all holomorphic functions s.t. $f(0) = 0$ and $f'(z) = f(z)g(z)$ for all $z \in U$
QUESTION [6 upvotes]: Let $U$ be a simply connected, open subset of $\Bbb C$ (not all of $\mathbb{C}$) containing $0$ and $1$. 
Given a holomorphic function $g: U \rightarrow \mathbb{C}$ with $g(0) \neq g(1)$, how do you find all holomorphic functions $f: U \rightarrow \mathbb{C}$ such that
$f(0) = 0$, and
$f'(z) = f(z)g(z)$ for all $z \in U$?

REPLY [2 votes]: The only such function is $f=0$. Because if $f$ does not vanish
identically then $f$ has a zero of some finite order $n$ at the
origin. Hence $f'$ has a zero of order $n-1$. But $fg$ has a zero
of order at least $n$, so $f'=fg$ is impossible.