Liouville theorem

Every bounded entire function is constant.

The image of every nonconstant entire function is dense in $\mathbb{C}$. If a disc about $w$ were omitted, then $1/(f-w)$ would be bounded and entire, so Liouville's theorem would make $f$ constant.

If an entire function satisfies $|h(z)|\le |\mathrm{Re}z{|}^{-1/2}$ away from the imaginary axis, then $h=0$. Cauchy's formula on $|z|=R$ bounds every Taylor coefficient by a constant timeswhich tends to zero as $R\to \infty$.

An entire function bounded by a polynomial in the radius is itself a polynomial, by Cauchy estimates.