I am trying to solve a problem from complex analysis which is concerning normal families.The problem is the following:
Show that,if $\mathcal F\subset \mathcal H(\Omega)$ is a normal family of holomorphic functions on $\Omega$,then $\mathcal F':=\{f':f\in \mathcal F\}\subset \mathcal H(\Omega)$ is again a normal family of holomorphic functions on $\Omega$.
I have come up with a solution but I am not sure whether I have made any mistake.
My Solution:
Let $f\in \mathcal F$ .Since $\mathcal F$ is uniformly bounded over compact subsets of $\Omega$(we say,uniformly bounded on compacta,so we can equivalently say $\mathcal F$ is locally uniformly bounded in $\Omega$.Now suppose $z_0\in \Omega$ and $r>0$ such that $\overline{B(z_0,r)}\subset \Omega$.It suffieces to show $\mathcal F'$ is uniformly bounded on $\overline{B(z_0,r)}$.Use Cauchy's integral formula for first order derivative, $f'(z_0)=\frac{1}{2\pi i}\int_{C_{z_0}(r)} \frac{f(w)}{(w-z_0)^2}dw$ and hence we can estimate $|f'(z_0)|\leq \frac{M}{r}$ for all $f\in \mathcal F$ .We are using the uniform bound $M$ on $\mathcal F$.Since $r>0$ is independent of $f$.Hence we are we have shown that $\mathcal F'$ is uniformly bounded on the compact disc $\overline{ B(z_0,r)}$ and hence also on $B(z_0,r)$ and hence we can say that the derived family $\mathcal F'$ is uniformly bounded on compacta in $\Omega$.Hence the proof.
I just want to cross-verify if it is flawless.A little help from the stack exchange folks is highly appreciated.