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Let $G$ be an open connected set and let $a \in G$. For $M > 0$, let $F_{M}$ denote the family of holomorphic functions in $G$ with the property $|f(a)| \leq M \,$ and $\Re(f) >0$ for all $z \in G$. Show that the family is normal.

Attempt: Since, $\Re(f) >0$, we can consider the family $G_M:=\{\frac{f(z)-1}{f(z)+1}\, |\, f \in F_M\}$. Then as $T(z)=\frac{z-1}{z+1}$ maps $\Re(z) >0$ to the unit disk, we have $G_M$ is locally bounded and hence normal.

So, if $\{f_n\}$ is any sequence in $F_M$, there exists a subsequence $f_{n_k}$ such that $g_{n_k} \to g$ on compact subsets of the unit disk.

Can I just apply $T^{-1}$ on both sides to conclude? I feel like I am missing something since I have not used $|f(a)| \leq M $ anywhere.

divergenceball
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  • Might be relevant here – approximation Aug 28 '22 at 02:58
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    Normality for $g_n$ may allow convergence to the constant $1$ which would give divergence to infinity for the corresponding $f_n$; show that the boundness condition on $f_n(a)$ precludes that so you are good – Conrad Aug 28 '22 at 03:03

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