View source for Equilibrium relation

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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

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

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 [[#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====
<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">  H. Wittich,   "Neueste Ergebnisse über eindeutige analytische Funktionen" , Springer  (1955)</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">  B.V. Shabat,   "Introduction of complex analysis" , '''2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[5]</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></table>



====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|Nevanlinna theorems]] and [[Value-distribution theory|Value-distribution theory]] for the notions of counting function and proximity function.

====References====
<table><TR><TD valign="top">[a1]</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>