Schwarz Lemma Proof

Question

I'm Using Function theory on one complex variable by Robert. E . green

In the proof of the Schwarz' lemma, they have used the function $g(z)=\frac{f(z)}{z}$ for all non zero $z\in\mathbb{D}$. And then the Riemann removable singularity theorem have been used. But for this don't we need the bounded property of $g(z)$ ? if so , how to prove that $g(z)$ is bounded on $\mathbb{D}$

What I have noticed is that before proving the Riemann's removable singularity theorem we cannot say that $g(z)$ is holomorphic, thus the maximum modulus principle too can NOT be used.

Answer 1

$f$ is complex differentiable in $\Bbb D$ with $f(0) = 0$, therefore $$ g(z) = \frac{f(z)}{z} = \frac{f(z)-f(0)}{z - 0} \to f'(0) \text{ for } z \to 0 $$ So $\lim_{z \to 0}g(z)$ exists, and then Riemann's removable singularity theorem implies that $g$ has a removable singularity at $z=0$.

Or even simpler: $$ \lim_{z \to 0} \, (z-0) \, g(z) = \lim_{z \to 0} \, f(z) = 0 $$ so criterion #4 in Riemann's theorem about removable singularities is satisfied for $g$.

Answer 2

I was able to find the answer, Thanks to the comment of @Martin R.
$\lim\limits_{z\to 0}g(z)=f^{'}(0)$ will imply that $\exists\delta$ such that
$$|z|<\delta\Rightarrow|g(z)-f^{'}(0)|<1\Rightarrow|g(z)|<1+|f^{'}(0)|$$

Also when $|z|\geq\delta$, $|g(z)|=\frac{|f(z)|}{|z|}\leq\frac{1}{|z|}\leq\frac{1}{\delta}$.
hence $g(z)$ is bounded.