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Montel's Great theorem states

Let $\mathcal{F}$ be a collection of analytic functions on a region $\Omega$ such that all of the $f\in \mathcal{F}$ omit the same two values $a,b$. Then the family is normal.

By Montel's theorem (a family of holomorphic functions on a region is normal iff it is locally uniformly bounded) the previous theorem can also be stated as:

Let $\mathcal{F}$ be a collection of analytic functions on a region $\Omega$ such that all of the $f\in \mathcal{F}$ omit the same two values $a,b$. Then the family is locally uniformly bounded.

So far, no problem;

My question is: isn't the family $\mathcal{F}:=\{f_a(z)=a;a\in \mathbb{C}-\{0,1\}\}$ a counterexample? The functions in the family are clearly analytic (since they are constants), and omit the values $0,1$, so $\mathcal{F}$ satisfies the hypotesis of M.G.T., but they are not locally uniformly bounded.

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It depends on which space of analytic functions you consider:

  1. The space of analytic functions $\Omega \to \Bbb C$ with convergence with respect to the euclidean metric. The limit functions are holomorphic in $\Omega$. A family is normal in this space if and only if it is locally uniformly bounded, this is Montel's theorem.

  2. The space of analytic functions $\Omega \to \hat{\Bbb C}$ with convergence with respect to the spherical metric. The limit functions are meromorphic in $\Omega$ or identically $\infty$. A family is normal in this space if and only if the spherical derivative is locally uniformly bounded, this is Marty's theorem.

The family of holomorphic functions omitting two fixed values is normal in the second space, i.e. with respect to the spherical metric. In your example all functions are constant, and the spherical derivative $$ \frac{2|f'(z)|}{1+|f(z)|^2} $$ is zero.

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  • $\begingroup$ Thus if I add the hypotesis $|f(z_0)|<K$ for all the functions in the family the G.M.T. becomes true even with the euclidean metric? $\endgroup$ – Caffeine Sep 12 '19 at 11:59
  • $\begingroup$ @GabrieleCassese: Yes. In that case all limit functions are holomorphic, and Montel's theorem applies. $\endgroup$ – Martin R Sep 12 '19 at 12:02

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