Equilibrium relation

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)