Proving an entire, complex function with below bounded modulus is constant.
The map $z\mapsto 1/f(z)$ is
- well defined;
- entire;
- bounded on the complex plane.
Here is a generalization.
If $|f| \geq 1$, then $|\frac{1}{f}| \leq 1$. Apply Liouville.
The map $z\mapsto 1/f(z)$ is
Here is a generalization.
If $|f| \geq 1$, then $|\frac{1}{f}| \leq 1$. Apply Liouville.