Entire doubly periodic functions are constant

Let \(\omega_1,\omega_2\) be two real-linearly independent periods of the entire function \(f\). Consider the closed fundamental parallelogram \[ P=\{t_1\omega_1+t_2\omega_2:0\le t_1,t_2\le1\}. \] It is compact, so continuity of \(f\) gives \[ M=\max_{z\in P}|f(z)|<\infty. \]

Every \(z\in\mathbb C\) can be written \[ z=z_0+m\omega_1+n\omega_2 \] with \(m,n\in\mathbb Z\) and \(z_0\in P\). By periodicity, \[ f(z)=f(z_0), \] so \(|f(z)|\le M\) for all \(z\in\mathbb C\). Thus \(f\) is bounded and entire; Liouville’s theorem implies that \(f\) is constant.