Bloch, Schottky, and the Picard Theorems — Examples

Fact [L1] gives the lower bound $\beta (f)\ge 1\mathrm{/}48$ for every normalized holomorphic disc map.

Applying step 1.1 to the present function is exactly the stated example.

Apply [L1] with $R=1\mathrm{/}2$. This gives a constant $C(1\mathrm{/}2,r)$ depending only on $r$.

The constant from step 1.1 works for the present function, so one may take $C_r:=C(1\mathrm{/}2,r)$.

Facts [L1] and [L2] show that $e^z$ is a nonconstant entire function whose omitted finite-value set is exactly $\{0\}$.

Fact [L3] says no nonconstant entire function can omit two finite values, while step 1.1 exhibits one omitting exactly one. Therefore the bound in Little Picard is sharp.

Since the exponential never vanishes, $f(z)$ never equals $0$ on the punctured disc.

Fix $w\mathrm{\ne }0$. By [L1] and [L2], choose $\lambda \in \mathbb{C}$ with $e^{\lambda }=w$; then every number $\lambda +2\pi ik$ is another logarithm of $w$. For every integer $k$ with $\lambda +2\pi ik\mathrm{\ne }0$, set $z_k:=1\mathrm{/}(\lambda +2\pi ik)$. At most one integer is excluded, while all the remaining $z_k$ satisfy $f(z_k)=w$ and $z_k\to 0$ as $\mathrm{\mid }k\mathrm{\mid }\to \mathrm{\infty }$. Thus every nonzero value occurs infinitely often near $0$.

Therefore Great Picard is sharp: one finite exceptional value can occur, namely $0$.

Fact [L1] makes $e^z$ meromorphic on $\mathbb{C}$ with no poles, and [L2] identifies its image as $\mathbb{C}\setminus \{0\}$.

As a sphere-valued map, the omitted values are therefore $0$ and $\mathrm{\infty }$. Since $e^z$ is nonconstant, this refutes the claim that at most one sphere value can be omitted.

Fact [L1] already states the omitted-value conclusion for every nonconstant entire function, with no boundedness hypothesis.

Therefore the asserted extra boundedness assumption is false.

Fact [L1] is already a nonconstant meromorphic function on $\mathbb{C}$ omitting two sphere values.

Therefore the claim "at most one sphere value" is false.