I am trying to do all the exercises from Tom M. Apostol's Dirichlet series and Modular functions in number theory and I could not think about the following problem from the chapter "Elliptic functions".
Problem - Let $\omega_1$ and $\omega_2$ be complex numbers with non real ratio. Let $f(z)$ be an entire function and assume there are constants $a$ and $b$ such that $f(z+\omega_1) = a f(z)$ , $f(z+ \omega_2) = b f(z)$ , for all $z$. Prove that $f(z) = A\exp(Bz)$, where $\exp(x) = e^x$ and $A$, $B$ are constants.
I tried a solution by putting $z= 0$ and seeing what happens for $\omega_1$ , $\omega_2$ but I couldn't prove any relation that could imply that $f$ must be an exponential function.
Can somebody please give hints.
Hint: consider the ratio $f/(A \exp(Bz))$ with $A$ and $B$ chosen such that $A \exp(Bz)$ has the same quasiperiods as $f$, i.e. the ratio is elliptic. Remember Liouville's theorem