Function of bounded characteristic

in a domain $ D $ of the complex plane $ \mathbf C $

A meromorphic function $ f ( z) $ in $ D $ that can be represented in $ D $ as the quotient of two bounded analytic functions,

$$ \tag{1 } f ( z) = \ \frac{g _ {1} ( z) }{g _ {2} ( z) } ,\ \ | g _ {1} |, | g _ {2} | \leq 1,\ \ z \in D , $$

is called a function of bounded type. The class most studied is the class $ N ( \Delta ) $ of functions of bounded type in the unit disc $ \Delta = \{ {z \in \mathbf C } : {| z | < 1 } \} $: A meromorphic function $ f ( z) $ in $ \Delta $ belongs to $ N ( \Delta ) $ if and only if its characteristic $ T ( r; f ) $ is bounded (Nevanlinna's theorem):

$$ \tag{2 } T ( r; f ) = \ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} ^ {+} \ | f ( re ^ {i \theta } ) | \ d \theta + $$

$$ + \sum \mathop{\rm ln} { \frac{r}{| b _ \nu | } } + \lambda \mathop{\rm ln} r \leq C ( f ) < \infty ,\ 0 < r < 1. $$

Here the sum on the right-hand side is taken over all poles $ b _ \nu $ of $ f ( z) $ with $ 0 < | b _ \nu | < r $, and each pole is taken as many times as its multiplicity; $ \lambda \geq 0 $ is the multiplicity of the pole at the origin. Hence functions in the class $ N ( \Delta ) $ are also called functions of bounded characteristic. The following sufficient condition is also of interest: If a meromorphic function $ f ( z) $ in $ \Delta $ does not take a set of values $ E $ of positive capacity, $ \mathop{\rm cap} E > 0 $, then $ f ( z) \in N ( \Delta ) $.

The functions $ f ( z) $ in $ N ( \Delta ) $ have the following properties: 1) $ f ( z) $ has angular boundary values $ f ( e ^ {i \theta } ) $, with $ \mathop{\rm ln} | f ( e ^ {i \theta } ) | \in L _ {1} ( \Gamma ) $, almost-everywhere on the unit circle $ \Gamma = \{ {z \in \mathbf C } : {| z | = 1 } \} $; 2) if $ f ( e ^ {i \theta } ) = 0 $ on a set of points of $ \Gamma $ of positive measure, then $ f ( z) \equiv 0 $; 3) a function $ f ( z) \in N ( \Delta ) $ is characterized by an integral representation of the form

$$ \tag{3 } f ( z) = \ z ^ {m} e ^ {i \lambda } \frac{B _ {1} ( z; a _ \mu ) }{B _ {2} ( z; b _ \nu ) } \times $$

$$ \times \mathop{\rm exp} \left \{ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} | f ( e ^ {i \theta } ) | \frac{e ^ {i \theta } + z }{e ^ {i \theta } - z } d \theta + { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \frac{e ^ {i \theta } + z }{e ^ {i \theta } - z } d \Phi ( \theta ) \right \} , $$

where $ m $ is the integer such that $ f ( z) = z ^ {m} \phi ( z) $, $ \phi ( 0) \neq 0, \infty $; $ \lambda $ is real; $ B _ {1} ( z; a _ \mu ) $ and $ B _ {2} ( z; b _ \nu ) $ are the Blaschke products taken over all zeros $ a _ \mu \neq 0 $ and poles $ b _ \nu \neq 0 $ of $ f ( z) $ inside $ \Delta $, counted with multiplicity (cf. Blaschke product); and $ \Phi ( \theta ) $ is a singular function of bounded variation on $ [ 0, 2 \pi ] $ with derivative equal to zero almost-everywhere.

The subclass $ N ^ {*} ( \Delta ) $ of $ N ( \Delta ) $ consisting of all holomorphic functions $ f ( z) $ in $ N ( \Delta ) $ is also of interest. A necessary and sufficient condition for a holomorphic function $ f ( z) $ to be in $ N ^ {*} ( \Delta ) $ is that it satisfies the following condition, deduced from (2),

$$ \tag{4 } { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} ^ {+} \ | f ( re ^ {i \theta } ) | \ d \theta \leq \ C ( f ) < \infty ,\ \ 0 < r < 1. $$

For $ f ( z) \in N ^ {*} ( \Delta ) $ one must have $ B _ {2} ( z; b _ \nu ) \equiv 1 $, $ m \geq 0 $ in (3).

Condition (4) is equivalent to the requirement that the subharmonic function $ \mathop{\rm ln} ^ {+} | f ( z) | $ has a harmonic majorant in the whole disc $ \Delta $. The condition in this form is usually taken to define the class $ N ^ {*} ( D) $ of holomorphic functions on arbitrary domains $ D \subset \mathbf C $: $ f ( z) \in N ^ {*} ( D) $ if and only if $ \mathop{\rm ln} ^ {+} | f ( z) | $ has a harmonic majorant in the whole domain $ D $.

Suppose that the function $ w = w ( z) $ realizes a conformal universal covering mapping $ \Delta \rightarrow D $( i.e. a single-valued analytic function on $ \Delta $ that is automorphic with respect to the group $ G $ of fractional-linear transformations of the disc $ \Delta $ onto itself corresponding to $ D $). Then $ f ( w) \in N ^ {*} ( D) $ if and only if the composite function $ f ( w ( z)) $ is automorphic relative to $ G $ and $ f ( w ( z)) \in N ^ {*} ( \Delta ) $. If $ D $ is a finitely-connected domain and if its boundary $ \partial D $ is rectifiable, then the angular boundary values $ f ( \zeta ) $, $ \zeta \in \partial D $, of $ f ( z) \in N ^ {*} ( D) $ exist almost-everywhere on $ \partial D $, and $ \mathop{\rm ln} | f ( \zeta ) | $ is summable with respect to harmonic measure on $ \partial D $( for more details see the survey [4]).

Now let $ f ( z) $, $ z = ( z _ {1} \dots z _ {n} ) $, $ n > 1 $, be a holomorphic function of several variables on the unit polydisc $ \Delta ^ {n} = \{ {z \in \mathbf C ^ {n} } : {| z _ {j} | < 1, j = 1 \dots n } \} $, and let $ T ^ {n} $ be the skeleton of $ \Delta ^ {n} $, $ T ^ {n} = \{ {z \in \mathbf C ^ {n} } : {| z _ {j} | = 1, j = 1 \dots n } \} $. The class $ N ^ {*} ( \Delta ^ {n} ) $ of functions of bounded characteristic is defined by a condition generalizing (4):

$$ \int\limits _ {T ^ {n} } \mathop{\rm ln} ^ {+} \ | f ( r \zeta ) | \ dm _ {n} ( \zeta ) \leq \ C ( f ) < \infty ,\ \ 0 < r < 1, $$

where $ \zeta = ( \zeta _ {1} \dots \zeta _ {n} ) \in T ^ {n} $ and $ m _ {n} ( \zeta ) $ is the normalized Haar measure on $ T ^ {n} $, $ m _ {n} ( T ^ {n} ) = 1 $. A holomorphic function $ f ( z) $ in the class $ N ^ {*} ( \Delta ^ {n} ) $ has radial boundary values $ \lim\limits _ {r \rightarrow 1 } f ( r \zeta ) = f ( \zeta ) $, $ \zeta \in T ^ {n} $, almost-everywhere on $ T ^ {n} $ with respect to Haar measure $ m _ {n} $, and $ \mathop{\rm ln} | f ( \zeta ) | $ is summable on $ T ^ {n} $. If the original definition (1) of a function of bounded type on $ D = \Delta ^ {n} $ is retained, then a function $ f ( z) $ of bounded type is a function of bounded characteristic, $ N ( \Delta ^ {n} ) \subset N ^ {*} ( \Delta ^ {n} ) $. However, for $ n > 1 $ there are functions $ g ( z) \in N ^ {*} ( \Delta ^ {n} ) $ 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 $ f \in N ( \Delta ) $ are sometimes called functions of bounded form or have no special name at all, the class $ N ^ {*} ( \Delta ) $ being more important.