Research proof index

Let $f$ be analytic on an open set $s$ containing $0$, and suppose $f(0)=0$. Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$. Then the apparent singularity at $0$ is removable and $g$ is analytic throughout $s$.

Suppose $R>0$ and $f$ is analytic on the closed disc $|z|\le R$ with $f(0)=0$. Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$. Then $g$ is analytic on the entire closed disc, including at the removed singularity.

A closed-disc form of the Borel–Carathéodory theorem. Let $R,M>0$, let $f$ be analytic on $|z|\le R$, assume $f(0)=0$, and suppose $\operatorname{Re}f(z)\le M$ throughout that disc. If $r<R$ and $|z|\le r$, then $|f(z)|\le \frac{2Mr}{R-r}.$