# 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)

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.