Consider an entire function $f$. What can be said about $\inf_{z\in\mathbb{C}}|f(z)|$?
My attempt:
Consider $B:=\overline{B}(0,r)$ with $r>0$. Then $$ \inf_{z\in\mathbb{C}}|f(z)|\le \inf_{z\in B}|f(z)|\overset{\mathrm{Weierstrass}}{=}\min_{z\in B}|f(z)| =\min_{z\in\partial B}|f(z)|.$$
If $f=C$ constant, then $\inf_{z\in\mathbb{C}}|f(z)|\le C$. Otherwise, $f$ is a nonconstant entire function, implying it is unbounded, i.e. there is no $K\in \mathbb{R}$ such that $\inf_{z\in\mathbb{C}}|f(z)|\le |f(z)|\le K$. Therefore $\inf_{z\in\mathbb{C}}|f(z)|=-\infty$. Or should I say that it doesn't exist?
Thanks.
If $f$ is constant and it always takes the value $\omega$, then $\inf_{z\in\mathbb C}\bigl\lvert f(z)\bigr\rvert=\lvert\omega\rvert$.
Otherwise, since $f(\mathbb C)$ is dense, $\inf_{z\in\mathbb C}\bigl\lvert f(z)\bigr\rvert=0$.