Normal family

of analytic functions in a domain

A family $ S $ of single-valued analytic functions $ f ( z) $ of complex variables $ z = ( z _ {1} \dots z _ {n} ) $ in a domain $ D $ in the space $ \mathbf C ^ {n} $, $ n \geq 1 $, such that from any sequence of functions in $ S $ one can extract a subsequence $ \{ f _ {v} ( z) \} $ that converges uniformly on compact subsets in $ D $ to an analytic function or to infinity. Uniform convergence to infinity on compact subsets means, by definition, that for any compact set $ K \subset D $ and any $ M > 0 $ one can find an $ N = N ( K, M) $ such that $ | f _ {v} ( z) | > M $ for all $ v > N $, $ z \in K $.

A family $ S $ is called a normal family at a point $ z ^ {0} \in D $ if $ S $ is normal in some ball with centre at $ z ^ {0} $. A family $ S $ is normal in $ D $ if and only if it is normal at every point $ z ^ {0} \in D $. Every compact family of holomorphic functions is normal; the converse conclusion is false (see Compactness principle). If a family $ S $ of holomorphic functions in a domain $ D \subset \mathbf C ^ {n} $ has the property that all functions $ f ( z) \in S $ omit two fixed values, then $ S $ is normal in $ D $( Montel's theorem). This criterion of normality considerably simplifies the investigation of analytic functions in a neighbourhood of an essential singular point (see also Picard theorem).

A normal family of meromorphic functions in a domain $ D \subset \mathbf C = \mathbf C ^ {1} $ is defined similarly: A family $ S $ of meromorphic functions in $ D $ is normal if from every sequence of functions in $ S $ one can extract a subsequence $ \{ f _ {v} ( z) \} $ that converges uniformly on compact subsets in $ D $ to a meromorphic function or to infinity. By definition, $ \{ f _ {v} ( z) \} $ converges uniformly on compact subsets in $ D $ to $ f ( z) $( the case $ f ( z) \equiv \infty $ is excluded) if for any compact set $ K \subset D $ and any $ \epsilon > 0 $ there is an $ N = N ( \epsilon , K) $ and a disc $ B = B ( z ^ {0} , r) $ of radius $ r = r ( \epsilon , K) $ with centre at some point $ z ^ {0} \in K $ such that for $ v > N $,

$$ | f _ {v} ( z) - f ( z) | < \epsilon ,\ \ z \in B, $$

when $ f ( z ^ {0} ) \neq \infty $, or

$$ \left | \frac{1}{f _ {v} ( z) } - { \frac{1}{f ( z) } } \right | < \epsilon ,\ \ z \in B, $$

when $ f ( z ^ {0} ) = \infty $. If a family $ S $ of meromorphic functions in a domain $ D \subset \mathbf C $ has the property that all functions $ f \in S $ omit three fixed values, then $ S $ is normal (Montel's theorem). A family $ S $ of meromorphic functions is normal in a domain $ D \subset \mathbf C $ if and only if

$$ \sup \{ {\rho ( f ( z)) } : {f \in S } \} < \infty $$

on every compact set $ K \subset D $, where

$$ \rho ( f ( z)) = \ \frac{| f ^ { \prime } ( z) | }{1 + | f ( z) | ^ {2} } $$

is the so-called spherical derivative of $ f ( z) $.

From the 1930s onwards great value was attached to the study of boundary properties of analytic functions (see also Cluster set, [3], [4]). A meromorphic function $ f ( z) $ in a simply-connected domain $ D \subset \mathbf C $ is said to be a normal function in the domain $ D $ if the family $ \{ f ( \gamma ( z)) \} $ is normal in $ D $, where $ \gamma ( z) $ ranges over the family of all conformal automorphisms of $ D $. A function $ f ( z) $ is called normal in a multiply-connected domain $ D $ if it is normal on the universal covering surface of $ D $. If a meromorphic function $ f ( z) $ in $ D $ omits three values, then $ f ( z) $ is normal. For $ f ( z) $, $ f ( z) \neq \textrm{ const } $, to be normal in the unit disc $ G = \{ {z \in \mathbf C } : {| z | < 1 } \} $ it is necessary and sufficient that

$$ \frac{| f ^ { \prime } ( z) | }{1 + | f ( z) | ^ {2} } < \ \frac{c}{1 - | z | ^ {2} } ,\ \ z \in G,\ \ c = c ( f ) = \textrm{ const } . $$

For a normal meromorphic function $ f ( z) $ in the unit disc $ G $ the existence of an asymptotic value $ \alpha $ at a boundary point $ \zeta \in \Gamma = \{ {\zeta \in \mathbf C } : {| \zeta | = 1 } \} $ implies that $ \alpha $ is a non-tangential boundary value (cf. Angular boundary value) of $ f ( z) $ at $ \zeta $. However, a meromorphic normal function in $ G $ need not have asymptotic values at all. On the other hand, if $ f ( z) $ is a holomorphic normal function in $ G $, then non-tangential boundary values exist even on a set of points of the unit circle $ \Gamma $ that is dense in $ \Gamma $.

References

[1]	P. Montel, "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars (1927)
[2]	A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian)
[3]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6
[4]	A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)

Comments

Let $ D _ {1} \subseteq \mathbf C ^ {m} $, $ D _ {2} \subseteq \mathbf C ^ {n} $ be domains. A family $ F $ of analytic mappings from $ D _ {1} $ to $ D _ {2} $ is called normal if from any sequence of mappings in $ F $ one can either extract a subsequence $ \{ f _ \nu ( z) \} $ that is uniformly convergent on compact subsets in $ D _ {1} $ to an analytic mapping from $ D _ {1} $ to $ D _ {2} $, or a subsequence $ \{ f _ \nu ( z) \} $ with the property that for every compact sets $ K _ {1} \subset D _ {1} $, $ K _ {2} \subset D _ {2} $ there is an $ N $ such that $ f _ \nu ( K _ {1} ) \cap K _ {2} = \emptyset $ for $ \nu > N $, see [a1].

References

[a1]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982)
[a2]	O. Lehto, K.I. Virtanen, "Boundary behaviour and normal meromorphic functions" Acta Math. , 97 (1957) pp. 47–65