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)

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].