Equilibrium relation
A relation expressing the connection between the growth of a function that is meromorphic for
, and its value distribution (see Value-distribution theory). Each meromorphic function
has the following equilibrium property: The sum of its counting function
, which characterizes the density of the distribution of
-points of
, and the proximity function
, which characterizes the average rate of approximation of
to the given number
, is invariant for different values of
. The equilibrium relation becomes more effective when using the spherical metric.
Let
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denote the spherical distance between two numbers and
, and, for each complex number
, let
![]() |
where
![]() |
and let denote the multiplicity of
-points of
for
. As
the function
differs from the Nevanlinna proximity function
by a bounded term. Therefore, on a circle
, the function
, as before, characterizes the average rate of approximation of
to
. The following result holds. For each value
,
, for any complex number
in the extended complex plane and for an arbitrary function
that is meromorphic in
, the equality (the equilibrium relation)
![]() |
holds, where
![]() |
and denotes the number of
-points of
in the disc
.
After the foundational work of R. Nevanlinna [1], the equilibrium relation was carried over to -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 -point of a function
is a point
such that
.
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" .
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) |
Equilibrium relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equilibrium_relation&oldid=40766