$f$ is an entire function if $|f(z)|\geq e^\pi $, what can we say about $f$

Question

$$|f(z)|\geq e^\pi, \frac{1}{|f(z)|}\leq \frac{1}{e^\pi}$$If $\frac{1}{|f(z)|}$ is entire we can conclude $f$ is constant, but is it entire?

Answer 1

Since $f$ is entire and it is never equal to $0$, $\frac1f$ is entire, and therefore you can deduce that it is constant. So, $f$ is constant.