If a holomorphic function on the complex plane has two periods which are linearly independent over $\mathbb R$ then is the function constant?

Question

Let $f: \mathbb C \to \mathbb C$ be holomorphic such that there exist $a,b \in \mathbb C$ , linearly independent over $\mathbb R$ such that $f(z)=f(z+a)=f(z+b) , \forall z \in \mathbb C$ . Then is it true that $f$ is constant ?

Answer 1

HINT: show that the function $f$ is bounded, since it is entire you can conclude that $f$ is constant by Liouville theorem.