A question about normal families

Question

This is Ahlfors q. 1, p. 227. Prove that in any region the family of analytic functions with positive real part is normal. Under what added condition is it locally bounded? Hint: Consider $e^{-f}$. Well I proved that $e^{-f}$ is normal, but I don't know what to do next, moreover If I am right normality implies locally boundeness in the analytic case,I'm trying to figure out why asked that question...

Answer 1

Yes, I am confused as to what Ahlfors meant. Montel's theorem says that a family of functions on a domain is normal if and only if it is locally bounded. I'm sure you know the direction "locally bounded implies normal", for the converse direction you just note that if $\mathcal{F}$ is a non-locally bounded family, then on a compact subset $K\subseteq\Omega$, and so there exists a sequence of functions $\{f_n\}$ with $f_n\in\mathcal{F}$ and a sequence of points $\{z_n\}$ in $\Omega$ such that $f_n(z_n)\to\infty$. Clearly $\{f_n\}$ can't have a subsequence which converges uniformly on $K$, and thus $\mathcal{F}$ can't be normal.