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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...

user1337
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User123456
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1 Answers1

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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.

Alex Youcis
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  • Thanks for the comment, but the relevant question is the first one,namely,prove that the family of functions of positive real part is bounded. – User123456 Mar 01 '13 at 05:43
  • @Alex I believe the author means "normal in the classical sense", where a uniform convergence to $\infty$ is also valid. Otherwise the sequence $f_n(z)=n$ has positive real part, and contradicts the normality of the family. – user1337 Sep 12 '13 at 14:20