Function of bounded characteristic
in a domain of the complex plane
A meromorphic function in
that can be represented in
as the quotient of two bounded analytic functions,
![]() | (1) |
is called a function of bounded type. The class most studied is the class of functions of bounded type in the unit disc
: A meromorphic function
in
belongs to
if and only if its characteristic
is bounded (Nevanlinna's theorem):
![]() | (2) |
![]() |
Here the sum on the right-hand side is taken over all poles of
with
, and each pole is taken as many times as its multiplicity;
is the multiplicity of the pole at the origin. Hence functions in the class
are also called functions of bounded characteristic. The following sufficient condition is also of interest: If a meromorphic function
in
does not take a set of values
of positive capacity,
, then
.
The functions in
have the following properties: 1)
has angular boundary values
, with
, almost-everywhere on the unit circle
; 2) if
on a set of points of
of positive measure, then
; 3) a function
is characterized by an integral representation of the form
![]() | (3) |
![]() |
where is the integer such that
,
;
is real;
and
are the Blaschke products taken over all zeros
and poles
of
inside
, counted with multiplicity (cf. Blaschke product); and
is a singular function of bounded variation on
with derivative equal to zero almost-everywhere.
The subclass of
consisting of all holomorphic functions
in
is also of interest. A necessary and sufficient condition for a holomorphic function
to be in
is that it satisfies the following condition, deduced from (2),
![]() | (4) |
For one must have
,
in (3).
Condition (4) is equivalent to the requirement that the subharmonic function has a harmonic majorant in the whole disc
. The condition in this form is usually taken to define the class
of holomorphic functions on arbitrary domains
:
if and only if
has a harmonic majorant in the whole domain
.
Suppose that the function realizes a conformal universal covering mapping
(i.e. a single-valued analytic function on
that is automorphic with respect to the group
of fractional-linear transformations of the disc
onto itself corresponding to
). Then
if and only if the composite function
is automorphic relative to
and
. If
is a finitely-connected domain and if its boundary
is rectifiable, then the angular boundary values
,
, of
exist almost-everywhere on
, and
is summable with respect to harmonic measure on
(for more details see the survey [4]).
Now let ,
,
, be a holomorphic function of several variables on the unit polydisc
, and let
be the skeleton of
,
. The class
of functions of bounded characteristic is defined by a condition generalizing (4):
![]() |
where and
is the normalized Haar measure on
,
. A holomorphic function
in the class
has radial boundary values
,
, almost-everywhere on
with respect to Haar measure
, and
is summable on
. If the original definition (1) of a function of bounded type on
is retained, then a function
of bounded type is a function of bounded characteristic,
. However, for
there are functions
that are not representable as the quotient of two bounded holomorphic functions (see [5]).
References
[1] | R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German) |
[2] | A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) (Translated from Russian) |
[3] | I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |
[4] | Itogi Nauk. Mat. Anal. 1963 (1965) pp. 5–80 |
[5] | W. Rudin, "Function theory in polydiscs" , Benjamin (1969) |
Comments
One should not confuse the notion of "function of bounded type" as defined above with that of an entire function of bounded type. For this reason, functions are sometimes called functions of bounded form or have no special name at all, the class
being more important.
Function of bounded characteristic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_of_bounded_characteristic&oldid=47008