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A relation expressing the connection between the growth of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e0360201.png" /> that is meromorphic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e0360202.png" />, and its value distribution (see [[Value-distribution theory|Value-distribution theory]]). Each meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e0360203.png" /> has the following equilibrium property: The sum of its counting function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e0360204.png" />, which characterizes the density of the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e0360205.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e0360206.png" />, and the proximity function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e0360207.png" />, which characterizes the average rate of approximation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e0360208.png" /> to the given number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e0360209.png" />, is invariant for different values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602010.png" />. The equilibrium relation becomes more effective when using the spherical metric.
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A relation expressing the connection between the growth of a function $f(z)$ that is meromorphic for $|z|<R\leq\infty$, and its value distribution (see [[Value-distribution theory|Value-distribution theory]]). Each meromorphic function $f(z)$ has the following equilibrium property: The sum of its counting function $N(r,a,f)$, which characterizes the density of the distribution of $a$-points of $f(z)$, and the proximity function $m(r,a,f)$, which characterizes the average rate of approximation of $f(z)$ to the given number $a$, is invariant for different values of $a$. The equilibrium relation becomes more effective when using the spherical metric.
  
 
Let
 
Let
  
<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/e/e036/e036020/e03602011.png" /></td> </tr></table>
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$$[a,b]=\frac{|a-b|}{\sqrt{1+|a|^2}\cdot\sqrt{1+|b|^2}}$$
  
denote the spherical distance between two numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602013.png" />, and, for each complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602014.png" />, let
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denote the spherical distance between two numbers $a$ and $b$, and, for each complex number $a$, let
  
<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/e/e036/e036020/e03602015.png" /></td> </tr></table>
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$$m^0(r,a,f)=\frac1{2\pi}\int\limits_0^{2\pi}\ln\frac1{[f(re^{i\theta}),a]}d\theta-\alpha(a,f),$$
  
 
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/e/e036/e036020/e03602016.png" /></td> </tr></table>
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$$\alpha(a,f)=\lim_{z\to0}\ln\frac{|z|^n}{[f(z),a]},$$
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602017.png" /> denote the multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602018.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602019.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602020.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602021.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602022.png" /> differs from the Nevanlinna proximity function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602023.png" /> by a bounded term. Therefore, on a circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602024.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602025.png" />, as before, characterizes the average rate of approximation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602026.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602027.png" />. The following result holds. For each value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602029.png" />, for any complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602030.png" /> in the extended complex plane and for an arbitrary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602031.png" /> that is meromorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602032.png" />, the equality (the equilibrium relation)
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and let $n=n(0,a,f)$ denote the multiplicity of $a$-points of $f(z)$ for $z=0$. As $r\to R$ the function $m^0(r,a,f)$ differs from the Nevanlinna proximity function $m(r,a,f)$ by a bounded term. Therefore, on a circle $|z|=r<R$, the function $m^0(r,a,f)$, as before, characterizes the average rate of approximation of $f(z)$ to $a$. The following result holds. For each value $r$, $0\leq r<R$, for any complex number $a$ in the extended complex plane and for an arbitrary function $f(z)$ that is meromorphic in $|z|<R\leq\infty$, the equality (the equilibrium 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/e/e036/e036020/e03602033.png" /></td> </tr></table>
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$$m^0(r,a,f)+N(r,a,f)=m^0(r,\infty,f)+N(r,\infty,f)$$
  
 
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/e/e036/e036020/e03602034.png" /></td> </tr></table>
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$$N(r,a,f)=\int\limits_0^r[n(t,a,f)-n(0,a,f)]\frac{dt}t+n(0,a,f)\ln r$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602035.png" /> denotes the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602036.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602037.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602038.png" />.
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and $n(t,a,f)$ denotes the number of $a$-points of $f(z)$ in the disc $\lbrace z\colon|z|\leq t\rbrace$.
  
After the foundational work of R. Nevanlinna [[#References|[1]]], the equilibrium relation was carried over to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602039.png" />-dimensional entire curves (see [[#References|[3]]]) and to holomorphic mappings (see [[#References|[4]]], [[#References|[5]]]).
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After the foundational work of R. Nevanlinna [[#References|[1]]], the equilibrium relation was carried over to $p$-dimensional entire curves (see [[#References|[3]]]) and to holomorphic mappings (see [[#References|[4]]], [[#References|[5]]]).
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602041.png" />-point of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602042.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036020/e03602044.png" />.
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An $a$-point of a function $f$ is a point $z$ such that $f(z)=a$.
  
The equilibrium relation is often referred to as the  "Ahlfors–Shimizu version of Nevanlinna's first main theoremAhlfors–Shimizu version of Nevanlinna's first main theorem" .
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The equilibrium relation is often referred to as the  "Ahlfors–Shimizu version of Nevanlinna's first main theorem".
  
 
See also [[Nevanlinna theorems|Nevanlinna theorems]] and [[Value-distribution theory|Value-distribution theory]] for the notions of counting function and proximity function.
 
See also [[Nevanlinna theorems|Nevanlinna theorems]] and [[Value-distribution theory|Value-distribution theory]] for the notions of counting function and proximity function.

Latest revision as of 16:17, 1 April 2017

A relation expressing the connection between the growth of a function $f(z)$ that is meromorphic for $|z|<R\leq\infty$, and its value distribution (see Value-distribution theory). Each meromorphic function $f(z)$ has the following equilibrium property: The sum of its counting function $N(r,a,f)$, which characterizes the density of the distribution of $a$-points of $f(z)$, and the proximity function $m(r,a,f)$, which characterizes the average rate of approximation of $f(z)$ to the given number $a$, is invariant for different values of $a$. The equilibrium relation becomes more effective when using the spherical metric.

Let

$$[a,b]=\frac{|a-b|}{\sqrt{1+|a|^2}\cdot\sqrt{1+|b|^2}}$$

denote the spherical distance between two numbers $a$ and $b$, and, for each complex number $a$, let

$$m^0(r,a,f)=\frac1{2\pi}\int\limits_0^{2\pi}\ln\frac1{[f(re^{i\theta}),a]}d\theta-\alpha(a,f),$$

where

$$\alpha(a,f)=\lim_{z\to0}\ln\frac{|z|^n}{[f(z),a]},$$

and let $n=n(0,a,f)$ denote the multiplicity of $a$-points of $f(z)$ for $z=0$. As $r\to R$ the function $m^0(r,a,f)$ differs from the Nevanlinna proximity function $m(r,a,f)$ by a bounded term. Therefore, on a circle $|z|=r<R$, the function $m^0(r,a,f)$, as before, characterizes the average rate of approximation of $f(z)$ to $a$. The following result holds. For each value $r$, $0\leq r<R$, for any complex number $a$ in the extended complex plane and for an arbitrary function $f(z)$ that is meromorphic in $|z|<R\leq\infty$, the equality (the equilibrium relation)

$$m^0(r,a,f)+N(r,a,f)=m^0(r,\infty,f)+N(r,\infty,f)$$

holds, where

$$N(r,a,f)=\int\limits_0^r[n(t,a,f)-n(0,a,f)]\frac{dt}t+n(0,a,f)\ln r$$

and $n(t,a,f)$ denotes the number of $a$-points of $f(z)$ in the disc $\lbrace z\colon|z|\leq t\rbrace$.

After the foundational work of R. Nevanlinna [1], the equilibrium relation was carried over to $p$-dimensional entire curves (see [3]) and to holomorphic mappings (see [4], [5]).

References

[1] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[2] H. Wittich, "Neueste Ergebnisse über eindeutige analytische Funktionen" , Springer (1955)
[3] H. Weyl, "Meromorphic functions and analytic curves" , Princeton Univ. Press (1943)
[4] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)
[5] P. Griffiths, J. King, "Nevanlinna theory and holomorphic mappings between algebraic varieties" Acta Math. , 130 (1973) pp. 145–220


Comments

An $a$-point of a function $f$ is a point $z$ such that $f(z)=a$.

The equilibrium relation is often referred to as the "Ahlfors–Shimizu version of Nevanlinna's first main theorem".

See also Nevanlinna theorems and Value-distribution theory for the notions of counting function and proximity function.

References

[a1] 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:
Equilibrium relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equilibrium_relation&oldid=15385
This article was adapted from an original article by V.P. Petrenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article