If a function is $1$-periodic then it writes as a holomorphic function of $e^{2 \pi i z}$

I want to prove the following:

Suppose that $g \in H(\mathbb{H})$ (where $\mathbb{H}$ is the upper half-plane) and $g(z+1) = g(z)$ for all $z \in \mathbb{H}$. Then there exists a function $G \in H(\mathbb{D} \setminus \{0\})$ such that $g(z) = G(e^{2 \pi i z})$.

I don't know how to solve / start this. I thought that I can find a branch of logarithm, but in $\mathbb{D} \setminus \{0\}$ there isn't one...