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''of analytic functions in a domain''
 
''of analytic functions in a domain''
  
A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n0675101.png" /> of single-valued analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n0675102.png" /> of complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n0675103.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n0675104.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n0675105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n0675106.png" />, such that from any sequence of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n0675107.png" /> one can extract a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n0675108.png" /> that converges uniformly on compact subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n0675109.png" /> to an analytic function or to infinity. Uniform convergence to infinity on compact subsets means, by definition, that for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751010.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751011.png" /> one can find an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751015.png" />.
+
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|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|essential singular point]] (see also [[Picard theorem|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
  
A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751016.png" /> is called a normal family at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751017.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751018.png" /> is normal in some ball with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751019.png" />. A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751020.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751021.png" /> if and only if it is normal at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751022.png" />. Every compact family of holomorphic functions is normal; the converse conclusion is false (see [[Compactness principle|Compactness principle]]). If a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751023.png" /> of holomorphic functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751024.png" /> has the property that all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751025.png" /> omit two fixed values, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751026.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751027.png" /> (Montel's theorem). This criterion of normality considerably simplifies the investigation of analytic functions in a neighbourhood of an [[Essential singular point|essential singular point]] (see also [[Picard theorem|Picard theorem]]).
+
$$
 +
\left |
 +
\frac{1}{f _ {v} ( z) }
 +
- {
 +
\frac{1}{f ( z) }
 +
} \right |
 +
< \epsilon ,\ \
 +
z \in B,
 +
$$
  
A normal family of meromorphic functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751028.png" /> is defined similarly: A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751029.png" /> of meromorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751030.png" /> is normal if from every sequence of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751031.png" /> one can extract a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751032.png" /> that converges uniformly on compact subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751033.png" /> to a meromorphic function or to infinity. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751034.png" /> converges uniformly on compact subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751035.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751036.png" /> (the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751037.png" /> is excluded) if for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751038.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751039.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751040.png" /> and a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751041.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751042.png" /> with centre at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751043.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751044.png" />,
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751045.png" /></td> </tr></table>
+
$$
 +
\sup  \{ {\rho ( f ( z)) } : {f \in S } \}
 +
< \infty
 +
$$
  
when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751046.png" />, or
+
on every compact set  $  K \subset  D $,  
 +
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751047.png" /></td> </tr></table>
+
$$
 +
\rho ( f ( z))  = \
  
when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751048.png" />. If a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751049.png" /> of meromorphic functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751050.png" /> has the property that all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751051.png" /> omit three fixed values, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751052.png" /> is normal (Montel's theorem). A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751053.png" /> of meromorphic functions is normal in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751054.png" /> if and only if
+
\frac{| f ^ { \prime } ( z) | }{1 + | f ( z) |  ^ {2} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751055.png" /></td> </tr></table>
+
$$
  
on every compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751056.png" />, where
+
is the so-called spherical derivative of  $  f ( z) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751057.png" /></td> </tr></table>
+
From the 1930s onwards great value was attached to the study of [[Boundary properties of analytic functions|boundary properties of analytic functions]] (see also [[Cluster set|Cluster set]], [[#References|[3]]], [[#References|[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|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
  
is the so-called spherical derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751058.png" />.
+
$$
  
From the 1930s onwards great value was attached to the study of [[Boundary properties of analytic functions|boundary properties of analytic functions]] (see also [[Cluster set|Cluster set]], [[#References|[3]]], [[#References|[4]]]). A meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751059.png" /> in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751060.png" /> is said to be a normal function in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751061.png" /> if the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751062.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751064.png" /> ranges over the family of all conformal automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751065.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751066.png" /> is called normal in a multiply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751067.png" /> if it is normal on the [[Universal covering|universal covering]] surface of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751068.png" />. If a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751069.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751070.png" /> omits three values, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751071.png" /> is normal. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751073.png" />, to be normal in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751074.png" /> it is necessary and sufficient that
+
\frac{| f ^ { \prime } ( z) | }{1 + | f ( z) | ^ {2} }
 +
  < \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751075.png" /></td> </tr></table>
+
\frac{c}{1 - | z |  ^ {2} }
 +
,\ \
 +
z \in G,\ \
 +
c = c ( f  ) = \textrm{ const } .
 +
$$
  
For a normal meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751076.png" /> in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751077.png" /> the existence of an [[Asymptotic value|asymptotic value]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751078.png" /> at a boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751079.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751080.png" /> is a non-tangential boundary value (cf. [[Angular boundary value|Angular boundary value]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751081.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751082.png" />. However, a meromorphic normal function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751083.png" /> need not have asymptotic values at all. On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751084.png" /> is a holomorphic normal function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751085.png" />, then non-tangential boundary values exist even on a set of points of the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751086.png" /> that is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751087.png" />.
+
For a normal meromorphic function $  f ( z) $
 +
in the unit disc $  G $
 +
the existence of an [[Asymptotic value|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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Montel,  "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars  (1927)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Montel,  "Leçons sur les familles normales de fonctions analytiques et leurs applications" , Gauthier-Villars  (1927)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 1;6</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751089.png" /> be domains. A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751090.png" /> of analytic mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751091.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751092.png" /> is called normal if from any sequence of mappings in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751093.png" /> one can either extract a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751094.png" /> that is uniformly convergent on compact subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751095.png" /> to an analytic mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751096.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751097.png" />, or a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751098.png" /> with the property that for every compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n06751099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n067510100.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n067510101.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n067510102.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067510/n067510103.png" />, see [[#References|[a1]]].
+
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 [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.G. Krantz,  "Function theory of several complex variables" , Wiley  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Lehto,  K.I. Virtanen,  "Boundary behaviour and normal meromorphic functions"  ''Acta Math.'' , '''97'''  (1957)  pp. 47–65</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.G. Krantz,  "Function theory of several complex variables" , Wiley  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Lehto,  K.I. Virtanen,  "Boundary behaviour and normal meromorphic functions"  ''Acta Math.'' , '''97'''  (1957)  pp. 47–65</TD></TR></table>

Latest revision as of 08:03, 6 June 2020


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
How to Cite This Entry:
Normal family. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_family&oldid=48012
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article