# Function of bounded characteristic

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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)