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Two fundamental theorems, proved by R. Nevanlinna (see [[#References|[1]]], [[#References|[2]]]), that are basic for the theory of value distribution of meromorphic functions (see [[Value-distribution theory|Value-distribution theory]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n0664901.png" /> be a [[Meromorphic function|meromorphic function]] on a disc
+
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<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/n066/n066490/n0664902.png" /></td> </tr></table>
+
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 +
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n0664903.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n0664904.png" /> is meromorphic in the entire open complex plane. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n0664905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n0664906.png" />, the proximity function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n0664907.png" /> to a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n0664908.png" /> is defined by
+
Two fundamental theorems, proved by R. Nevanlinna (see [[#References|[1]]], [[#References|[2]]]), that are basic for the theory of value distribution of meromorphic functions (see [[Value-distribution theory|Value-distribution theory]]). Let  $  f ( z) $
 +
be a [[Meromorphic function|meromorphic function]] on a disc
  
<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/n066/n066490/n0664909.png" /></td> </tr></table>
+
$$
 +
K _ {R}  = \{ {z } : {| z | < R \leq  \infty } \}
 +
,
 +
$$
  
<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/n066/n066490/n06649010.png" /></td> </tr></table>
+
where  $  R = \infty $
 +
means that  $  f ( z) $
 +
is meromorphic in the entire open complex plane. For every  $  r $,
 +
0 \leq  r < R $,
 +
the proximity function of  $  f ( z) $
 +
to a number  $  a $
 +
is defined by
  
and the counting function of the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649011.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649012.png" /> by
+
$$
 +
m ( r, \infty , f  )  = \
  
<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/n066/n066490/n06649013.png" /></td> </tr></table>
+
\frac{1}{2 \pi }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649014.png" /> denotes the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649015.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649016.png" />, counting multiplicities, in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649017.png" />, i.e. the number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649018.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649019.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649022.png" />.
+
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
\mathop{\rm ln}  ^ {+}  | f ( re ^ {i \theta } ) |  d \theta ,
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649023.png" /> is called the Nevanlinna characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649024.png" />.
+
$$
 +
m ( r, a, f  )  = m \left ( r, \infty ,
 +
\frac{1}{f - a }
 +
\right ) ,\  a \neq \infty ,
 +
$$
  
Nevanlinna's first theorem. For any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649025.png" /> that is meromorphic on a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649026.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649028.png" />, and any complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649029.png" />,
+
and the counting function of the number of  $  a $-
 +
points of  $  f ( z) $
 +
by
  
<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/n066/n066490/n06649030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$
 +
N ( r, a, f  )  = \
 +
\int\limits _ { 0 } ^ { r }
 +
 
 +
\frac{n ( t, a, f  ) - n ( 0, a, f  ) }{t}
 +
\
 +
dt + n ( 0, a, f  )  \mathop{\rm ln}  r,
 +
$$
 +
 
 +
where  $  n ( t, a, f  ) $
 +
denotes the number of  $  a $-
 +
points of  $  f ( z) $,
 +
counting multiplicities, in the disc  $  \{ {z } : {| z | \leq  t } \} $,
 +
i.e. the number of elements of  $  f ^ { - 1 } ( a ) \cap \{ {z } : {| z | \leq  t } \} $,
 +
and  $  \mathop{\rm ln}  ^ {+}  x = \mathop{\rm ln}  x $
 +
for  $  x \geq  1 $,
 +
$  \mathop{\rm ln}  ^ {+}  x = 0 $
 +
for  $  0 \leq  x < 1 $.
 +
 
 +
The function  $  T ( r, f  ) = m ( r, \infty , f  ) + N ( r, \infty , f  ) $
 +
is called the Nevanlinna characteristic of  $  f ( z) $.
 +
 
 +
Nevanlinna's first theorem. For any function  $  f ( z) $
 +
that is meromorphic on a disc  $  K _ {R} $,
 +
for any  $  r $,
 +
$  0 \leq  r < R $,
 +
and any complex number  $  a $,
 +
 
 +
$$ \tag{1 }
 +
m ( r, a, f  ) + N ( r, a, f  )  = T ( r, f  ) + \phi ( r, a),
 +
$$
  
 
where
 
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/n066/n066490/n06649031.png" /></td> </tr></table>
+
$$
 +
| \phi ( r, a) |  \leq    \mathop{\rm ln}  ^ {+}  | a | + |  \mathop{\rm ln} |  c
 +
| | +  \mathop{\rm ln}  2.
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649032.png" /> denotes the first non-zero coefficient in the Laurent expansion about zero of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649033.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649034.png" />, and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649035.png" /> itself if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649036.png" />. Thus, for a function whose characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649037.png" /> increases without limit as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649038.png" />, the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649039.png" />, considered for different values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649040.png" />, is equal to the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649041.png" /> up to a bounded additive term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649042.png" />. In this sense, all values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649043.png" /> are equivalent for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649044.png" /> that is meromorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649045.png" />. For this reason, the theory of value distribution of meromorphic functions concerns itself with questions about the asymptotic behaviour of one term, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649046.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649047.png" />, in the invariant sum (1).
+
Here $  c $
 +
denotes the first non-zero coefficient in the Laurent expansion about zero of the function $  f ( z) - a $
 +
if $  f ( 0) = a \neq \infty $,  
 +
and of $  f ( z) $
 +
itself if $  f ( 0) = \infty $.  
 +
Thus, for a function whose characteristic $  T ( r, f  ) $
 +
increases without limit as $  r \rightarrow R $,  
 +
the sum $  m ( r, a, f  ) + N ( r, a, f  ) $,  
 +
considered for different values of $  a $,  
 +
is equal to the value $  T ( r, f  ) $
 +
up to a bounded additive term $  \phi ( r, a) $.  
 +
In this sense, all values $  a $
 +
are equivalent for any function $  f ( z) $
 +
that is meromorphic on $  K _ {R} $.  
 +
For this reason, the theory of value distribution of meromorphic functions concerns itself with questions about the asymptotic behaviour of one term, $  m ( r, a, f  ) $
 +
or $  N ( r, a, f  ) $,  
 +
in the invariant sum (1).
  
Nevanlinna's second theorem shows that, for almost all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649048.png" />, the principal role in the sum (1) is played by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649049.png" />. The statement of the theorem is as follows.
+
Nevanlinna's second theorem shows that, for almost all points $  a $,  
 +
the principal role in the sum (1) is played by $  N ( r, a, f  ) $.  
 +
The statement of the theorem is as follows.
  
For any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649050.png" /> that is meromorphic on a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649051.png" />, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649053.png" />, and any distinct numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649054.png" /> in the extended complex plane, the relation
+
For any function $  f ( z) $
 +
that is meromorphic on a disc $  K _ {R} = \{ {z } : {| z | < R \leq  \infty } \} $,  
 +
every $  q $,  
 +
$  q \geq  3 $,  
 +
and any distinct numbers $  \{ a _ {k} \} _ {k = 1 }  ^ {q} $
 +
in the extended complex plane, the relation
  
<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/n066/n066490/n06649055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\sum _ {k = 1 } ^ { q }  m ( r, a _ {k} , f  )  \leq  \
 +
2T ( r, f  ) - N _ {1} ( r, f  ) + S ( r, f  )
 +
$$
  
 
holds, where
 
holds, 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/n066/n066490/n06649056.png" /></td> </tr></table>
+
$$
 +
N _ {1} ( r, f  )  = N ( r , \infty , 1 / f ^ { \prime } ) + 2N
 +
( r, \infty , f  ) - N ( r, \infty , f ^ { \prime } ),
 +
$$
  
and the term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649057.png" /> has the following properties:
+
and the term $  S ( r, f  ) $
 +
has the following properties:
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649058.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649059.png" /> is meromorphic in the entire open complex plane, then
+
1) If $  R = \infty $,  
 +
i.e. if $  f ( z) $
 +
is meromorphic in the entire open complex plane, then
  
<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/n066/n066490/n06649060.png" /></td> </tr></table>
+
$$
 +
S ( r, f  )  = O (  \mathop{\rm ln} ( r \cdot T ( r, f  ))),
 +
$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649061.png" />, for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649062.png" /> with the possible exception of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649063.png" /> of finite total measure.
+
as $  r \rightarrow \infty $,  
 +
for all values of $  r $
 +
with the possible exception of a set $  E $
 +
of finite total measure.
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649064.png" />, then
+
2) If $  R < \infty $,  
 +
then
  
<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/n066/n066490/n06649065.png" /></td> </tr></table>
+
$$
 +
S ( r, f  )  = \
 +
O \left (  \mathop{\rm ln} \left (
 +
\frac{R}{R - r }
 +
T ( r, f  ) \right ) \right ) ,
 +
$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649066.png" />, for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649067.png" /> with the possible exception of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649068.png" /> for which
+
as $  r \rightarrow R $,  
 +
for all values of $  r $
 +
with the possible exception of a set $  E $
 +
for which
  
<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/n066/n066490/n06649069.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { E }
 +
\frac{dr}{R - r }
 +
  < \infty .
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649070.png" /> is non-decreasing with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649071.png" />, and therefore the right-hand term in (2) cannot increase as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649072.png" /> more rapidly than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649073.png" /> outside some exceptional set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649074.png" />.
+
The function $  N _ {1} ( r, f  ) $
 +
is non-decreasing with increasing $  r $,  
 +
and therefore the right-hand term in (2) cannot increase as $  r \rightarrow R $
 +
more rapidly than $  2T ( r, f  ) $
 +
outside some exceptional set $  E $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Nevanilinna,  "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Nevanlinna,  "Le théorème de Picard–Borel et la théorie des fonctions méromorphes" , Gauthier-Villars  (1929)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Weyl,  "Meromorphic functions and analytic curves" , Princeton Univ. Press  (1943)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Ahlfors,  "The theory of meromorphic curves"  ''Acta Soc. Sci. Fennica. Nova Ser. A'' , '''3''' :  4  (1941)  pp. 1–31</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Cartan,  "Sur les zéros des combinations linéares de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649075.png" /> fonctions holomorphes données"  ''Mathematica (Cluj)'' , '''7'''  (1933)  pp. 5–31</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Griffiths,  J. King,  "Nevanlinna theory and holomorphic mappings between algebraic varieties"  ''Acta Math.'' , '''130'''  (1973)  pp. 145–220</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.P. Petrenko,  "The growth of meromorphic functions" , Khar'kov  (1978)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Nevanilinna,  "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Nevanlinna,  "Le théorème de Picard–Borel et la théorie des fonctions méromorphes" , Gauthier-Villars  (1929)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Weyl,  "Meromorphic functions and analytic curves" , Princeton Univ. Press  (1943)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Ahlfors,  "The theory of meromorphic curves"  ''Acta Soc. Sci. Fennica. Nova Ser. A'' , '''3''' :  4  (1941)  pp. 1–31</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Cartan,  "Sur les zéros des combinations linéares de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066490/n06649075.png" /> fonctions holomorphes données"  ''Mathematica (Cluj)'' , '''7'''  (1933)  pp. 5–31</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Griffiths,  J. King,  "Nevanlinna theory and holomorphic mappings between algebraic varieties"  ''Acta Math.'' , '''130'''  (1973)  pp. 145–220</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.P. Petrenko,  "The growth of meromorphic functions" , Khar'kov  (1978)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.K. Hayman,  "Meromorphic functions" , Oxford Univ. Press  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.A. Griffiths,  "Entire holomorphic mappings in one and several complex variables" , ''Annals Math. Studies'' , '''85''' , Princeton Univ. Press  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.K. Hayman,  "Meromorphic functions" , Oxford Univ. Press  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.A. Griffiths,  "Entire holomorphic mappings in one and several complex variables" , ''Annals Math. Studies'' , '''85''' , Princeton Univ. Press  (1976)</TD></TR></table>

Latest revision as of 08:02, 6 June 2020


Two fundamental theorems, proved by R. Nevanlinna (see [1], [2]), that are basic for the theory of value distribution of meromorphic functions (see Value-distribution theory). Let $ f ( z) $ be a meromorphic function on a disc

$$ K _ {R} = \{ {z } : {| z | < R \leq \infty } \} , $$

where $ R = \infty $ means that $ f ( z) $ is meromorphic in the entire open complex plane. For every $ r $, $ 0 \leq r < R $, the proximity function of $ f ( z) $ to a number $ a $ is defined by

$$ m ( r, \infty , f ) = \ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} ^ {+} | f ( re ^ {i \theta } ) | d \theta , $$

$$ m ( r, a, f ) = m \left ( r, \infty , \frac{1}{f - a } \right ) ,\ a \neq \infty , $$

and the counting function of the number of $ a $- points of $ f ( z) $ by

$$ N ( r, a, f ) = \ \int\limits _ { 0 } ^ { r } \frac{n ( t, a, f ) - n ( 0, a, f ) }{t} \ dt + n ( 0, a, f ) \mathop{\rm ln} r, $$

where $ n ( t, a, f ) $ denotes the number of $ a $- points of $ f ( z) $, counting multiplicities, in the disc $ \{ {z } : {| z | \leq t } \} $, i.e. the number of elements of $ f ^ { - 1 } ( a ) \cap \{ {z } : {| z | \leq t } \} $, and $ \mathop{\rm ln} ^ {+} x = \mathop{\rm ln} x $ for $ x \geq 1 $, $ \mathop{\rm ln} ^ {+} x = 0 $ for $ 0 \leq x < 1 $.

The function $ T ( r, f ) = m ( r, \infty , f ) + N ( r, \infty , f ) $ is called the Nevanlinna characteristic of $ f ( z) $.

Nevanlinna's first theorem. For any function $ f ( z) $ that is meromorphic on a disc $ K _ {R} $, for any $ r $, $ 0 \leq r < R $, and any complex number $ a $,

$$ \tag{1 } m ( r, a, f ) + N ( r, a, f ) = T ( r, f ) + \phi ( r, a), $$

where

$$ | \phi ( r, a) | \leq \mathop{\rm ln} ^ {+} | a | + | \mathop{\rm ln} | c | | + \mathop{\rm ln} 2. $$

Here $ c $ denotes the first non-zero coefficient in the Laurent expansion about zero of the function $ f ( z) - a $ if $ f ( 0) = a \neq \infty $, and of $ f ( z) $ itself if $ f ( 0) = \infty $. Thus, for a function whose characteristic $ T ( r, f ) $ increases without limit as $ r \rightarrow R $, the sum $ m ( r, a, f ) + N ( r, a, f ) $, considered for different values of $ a $, is equal to the value $ T ( r, f ) $ up to a bounded additive term $ \phi ( r, a) $. In this sense, all values $ a $ are equivalent for any function $ f ( z) $ that is meromorphic on $ K _ {R} $. For this reason, the theory of value distribution of meromorphic functions concerns itself with questions about the asymptotic behaviour of one term, $ m ( r, a, f ) $ or $ N ( r, a, f ) $, in the invariant sum (1).

Nevanlinna's second theorem shows that, for almost all points $ a $, the principal role in the sum (1) is played by $ N ( r, a, f ) $. The statement of the theorem is as follows.

For any function $ f ( z) $ that is meromorphic on a disc $ K _ {R} = \{ {z } : {| z | < R \leq \infty } \} $, every $ q $, $ q \geq 3 $, and any distinct numbers $ \{ a _ {k} \} _ {k = 1 } ^ {q} $ in the extended complex plane, the relation

$$ \tag{2 } \sum _ {k = 1 } ^ { q } m ( r, a _ {k} , f ) \leq \ 2T ( r, f ) - N _ {1} ( r, f ) + S ( r, f ) $$

holds, where

$$ N _ {1} ( r, f ) = N ( r , \infty , 1 / f ^ { \prime } ) + 2N ( r, \infty , f ) - N ( r, \infty , f ^ { \prime } ), $$

and the term $ S ( r, f ) $ has the following properties:

1) If $ R = \infty $, i.e. if $ f ( z) $ is meromorphic in the entire open complex plane, then

$$ S ( r, f ) = O ( \mathop{\rm ln} ( r \cdot T ( r, f ))), $$

as $ r \rightarrow \infty $, for all values of $ r $ with the possible exception of a set $ E $ of finite total measure.

2) If $ R < \infty $, then

$$ S ( r, f ) = \ O \left ( \mathop{\rm ln} \left ( \frac{R}{R - r } T ( r, f ) \right ) \right ) , $$

as $ r \rightarrow R $, for all values of $ r $ with the possible exception of a set $ E $ for which

$$ \int\limits _ { E } \frac{dr}{R - r } < \infty . $$

The function $ N _ {1} ( r, f ) $ is non-decreasing with increasing $ r $, and therefore the right-hand term in (2) cannot increase as $ r \rightarrow R $ more rapidly than $ 2T ( r, f ) $ outside some exceptional set $ E $.

References

[1] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[2] R. Nevanlinna, "Le théorème de Picard–Borel et la théorie des fonctions méromorphes" , Gauthier-Villars (1929)
[3] H. Weyl, "Meromorphic functions and analytic curves" , Princeton Univ. Press (1943)
[4] L. Ahlfors, "The theory of meromorphic curves" Acta Soc. Sci. Fennica. Nova Ser. A , 3 : 4 (1941) pp. 1–31
[5] H. Cartan, "Sur les zéros des combinations linéares de fonctions holomorphes données" Mathematica (Cluj) , 7 (1933) pp. 5–31
[6] P. Griffiths, J. King, "Nevanlinna theory and holomorphic mappings between algebraic varieties" Acta Math. , 130 (1973) pp. 145–220
[7] V.P. Petrenko, "The growth of meromorphic functions" , Khar'kov (1978) (In Russian)

Comments

References

[a1] W.K. Hayman, "Meromorphic functions" , Oxford Univ. Press (1964)
[a2] P.A. Griffiths, "Entire holomorphic mappings in one and several complex variables" , Annals Math. Studies , 85 , Princeton Univ. Press (1976)
How to Cite This Entry:
Nevanlinna theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nevanlinna_theorems&oldid=47963
This article was adapted from an original article by V.P. Petrenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article