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I'm reviewing Complex Analysis and I don't quite understand the concept of normal family. There is an exercise in Ahlfors' Complex Analysis: Prove that in any region the family of analytic functions with positive real part is normal.

I believe this is wrong. Theorem 15 says a family of analytic functions is normal with respect to $\mathbb{C}$ iff the functions are uniformly bounded on every compact set. However, this set contains the constant functions with arbitrary real part. So there is no way those functions are locally bounded.

Where am I wrong?

T C
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  • it gives me an answer, yes, but I'm not sure why my argument is wrong – T C May 27 '20 at 05:35
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    From the other question: “That is, every sequence of functions in the family has a subsequence which converges uniformly on compact subsets or tends uniformly to $\infty$ on compact subsets.” – Martin R May 27 '20 at 05:37

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In the same section Ahlfors gives a modified definition ("Classical definition") of normal families. A family is normal if every sequence in it has a subsequence converging uniformly on compacts sets or a subsequence tending to $\infty$ uniformly on compacts sets .Your example fails with this definition.

Kavi Rama Murthy
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