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Properties of analytic functions that are displayed as the function approaches the boundary of its domain of definition.
 
Properties of analytic functions that are displayed as the function approaches the boundary of its domain of definition.
  
 
It can be said that the study of boundary properties of analytic functions, understood in the widest sense of the word, began with the [[Sokhotskii theorem|Sokhotskii theorem]] and the [[Picard theorem|Picard theorem]] about the behaviour of analytic functions in a neighbourhood of isolated essential singular points (cf. [[Essential singular point|Essential singular point]]), which were obtained in the second half of the 19th century. The terms relevant to this approach to the study of boundary properties of analytic functions — which is now called the theory of prime ends and cluster sets (cf. [[Limit elements|Limit elements]]) — first appeared in a course given by P. Painlevé in 1895. The dissertation of P. Fatou (1906) is the first systematic study of certain boundary properties of analytic functions in a neighbourhood of the continuous boundary of their domain of definition. The theory of boundary properties made considerable advances in the first third of the 20th century, owing to the work of several scientists; it resumed its rapid advance in the second half of that century, accompanied by the appearance of new ideas and methods, novel directions and objects of study. Its development is closely connected with various fields of mathematical analysis and mathematics in general, first and foremost with probability theory, the theory of harmonic functions, the theory of conformal mapping, boundary value problems of analytic function theory, potential theory, value-distribution theory, Riemann surfaces, subharmonic functions and function algebras. The theory of boundary properties of analytic functions is closely connected with various fields of application of mathematics by way of boundary value problems.
 
It can be said that the study of boundary properties of analytic functions, understood in the widest sense of the word, began with the [[Sokhotskii theorem|Sokhotskii theorem]] and the [[Picard theorem|Picard theorem]] about the behaviour of analytic functions in a neighbourhood of isolated essential singular points (cf. [[Essential singular point|Essential singular point]]), which were obtained in the second half of the 19th century. The terms relevant to this approach to the study of boundary properties of analytic functions — which is now called the theory of prime ends and cluster sets (cf. [[Limit elements|Limit elements]]) — first appeared in a course given by P. Painlevé in 1895. The dissertation of P. Fatou (1906) is the first systematic study of certain boundary properties of analytic functions in a neighbourhood of the continuous boundary of their domain of definition. The theory of boundary properties made considerable advances in the first third of the 20th century, owing to the work of several scientists; it resumed its rapid advance in the second half of that century, accompanied by the appearance of new ideas and methods, novel directions and objects of study. Its development is closely connected with various fields of mathematical analysis and mathematics in general, first and foremost with probability theory, the theory of harmonic functions, the theory of conformal mapping, boundary value problems of analytic function theory, potential theory, value-distribution theory, Riemann surfaces, subharmonic functions and function algebras. The theory of boundary properties of analytic functions is closely connected with various fields of application of mathematics by way of boundary value problems.
  
Since the study of boundary properties is connected, in the first place, with the geometry of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b0173301.png" /> of the domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b0173302.png" /> of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b0173303.png" /> in one complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b0173304.png" />, three main approaches can be distinguished in the theory of boundary properties of analytic functions.
+
Since the study of boundary properties is connected, in the first place, with the geometry of the boundary $  \Gamma $
 +
of the domain of definition $  D $
 +
of an analytic function $  f(z) $
 +
in one complex variable $  z $,  
 +
three main approaches can be distinguished in the theory of boundary properties of analytic functions.
  
a) The study of the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b0173305.png" /> in a neighbourhood of an isolated boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b0173306.png" />. The most important case is that of an essential singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b0173307.png" />, which is dealt with in the theorems of Sokhotskii, Picard, Julia, and Iversen (cf. [[Sokhotskii theorem|Sokhotskii theorem]]; [[Picard theorem|Picard theorem]]; [[Julia theorem|Julia theorem]]; [[Iversen theorem|Iversen theorem]]).
+
a) The study of the behaviour of $  f(z) $
 +
in a neighbourhood of an isolated boundary point $  a \in \Gamma $.  
 +
The most important case is that of an essential singular point $  a $,  
 +
which is dealt with in the theorems of Sokhotskii, Picard, Julia, and Iversen (cf. [[Sokhotskii theorem|Sokhotskii theorem]]; [[Picard theorem|Picard theorem]]; [[Julia theorem|Julia theorem]]; [[Iversen theorem|Iversen theorem]]).
  
b) The study of the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b0173308.png" /> in the case when the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b0173309.png" /> is an everywhere-discontinuous set. V.V. Golubev's dissertation Single-valued analytic functions with a perfect set of singular points (1916, cf. [[#References|[1]]]) was of great importance in this connection.
+
b) The study of the behaviour of $  f(z) $
 +
in the case when the boundary $  \Gamma $
 +
is an everywhere-discontinuous set. V.V. Golubev's dissertation Single-valued analytic functions with a perfect set of singular points (1916, cf. [[#References|[1]]]) was of great importance in this connection.
  
c) The study of the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733010.png" /> when the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733011.png" /> is bounded by a continuous closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733012.png" /> and, in particular, by the unit circle.
+
c) The study of the behaviour of $  f(z) $
 +
when the domain $  D $
 +
is bounded by a continuous closed curve $  \Gamma $
 +
and, in particular, by the unit circle.
  
 
Cases a) and c) are, in a sense, extreme cases, while case b) is intermediate. Case c), which is discussed below, has been the subject of most intense study.
 
Cases a) and c) are, in a sense, extreme cases, while case b) is intermediate. Case c), which is discussed below, has been the subject of most intense study.
  
Let an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733013.png" /> be defined in a finite simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733014.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733015.png" />-plane bounded by a rectifiable Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733016.png" />. The following problems are fundamental in the classical approach to the study of boundary properties of analytic functions.
+
Let an analytic function $  f(z) $
 +
be defined in a finite simply-connected domain $  D $
 +
of the complex $  z $-
 +
plane bounded by a rectifiable Jordan curve $  \Gamma $.  
 +
The following problems are fundamental in the classical approach to the study of boundary properties of analytic functions.
 +
 
 +
1) The problem of the existence of boundary values, i.e. the question under which conditions and in which sense the boundary values of  $  f(z) $
 +
exist as the point  $  z $
 +
approaches  $  \Gamma $.
 +
This problem, as well as the succeeding ones, can be formulated in a different manner as the problem of identifying sufficiently extensive classes of analytic functions in  $  D $
 +
that have, in some sense, boundary values for sufficiently large sets of points on  $  \Gamma $.
 +
 
 +
2) The problem of boundary representation of  $  f(z) $,
 +
i.e. the problem under which conditions and with what kind of analytic apparatus the dependence of  $  f(z) $
 +
on its boundary values on  $  \Gamma $
 +
can be expressed. Clearly, this apparatus will be different for different classes of analytic functions.
 +
 
 +
3) The uniqueness problem, or the problem of the properties that a set  $  E \subset  \Gamma $
 +
should have such that two analytic functions of a given class coincide in  $  D $
 +
if their boundary values on  $  E $
 +
are identical.
 +
 
 +
The first result in the solution of the existence problem is the theorem of Fatou (1906): If an analytic function is bounded in the unit disc  $  D = \{ {z } : {| z | < 1 } \} $,
 +
$  | f(z) | \leq  M $,
 +
then radial boundary, or limit, values  $  f(e ^ {i \theta } ) = \lim\limits _ {r \rightarrow 1 - 0 }  f (re ^ {i \theta } ) $
 +
exist almost everywhere with respect to the Lebesgue measure on the unit circle  $  \Gamma = \{ {z } : {| z | = 1 } \} $.
 +
It can be shown that, under these conditions, not only radial, but also angular boundary values, or boundary values along all non-tangential paths, exist almost everywhere on  $  \Gamma $.
 +
This means that, for almost-all points  $  e ^ {i \theta } \in \Gamma $,
 +
$  f(z) $
 +
tends to a definite limit  $  f ( e ^ {i \theta } ) $
 +
as  $  z $
 +
tends to the point  $  e ^ {i \theta } $
 +
while remaining within an arbitrary fixed angle
  
1) The problem of the existence of boundary values, i.e. the question under which conditions and in which sense the boundary values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733017.png" /> exist as the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733018.png" /> approaches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733019.png" />. This problem, as well as the succeeding ones, can be formulated in a different manner as the problem of identifying sufficiently extensive classes of analytic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733020.png" /> that have, in some sense, boundary values for sufficiently large sets of points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733021.png" />.
+
$$
 +
\Delta (e ^ {i \theta } , \epsilon ) = \
 +
\{ | z | < 1 \} \cap
 +
\left \{ |  \mathop{\rm arg} (e ^ {i \theta } - z) | <
 +
{
 +
\frac \pi {2}
 +
} - \epsilon \right \} ,
 +
$$
  
2) The problem of boundary representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733022.png" />, i.e. the problem under which conditions and with what kind of analytic apparatus the dependence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733023.png" /> on its boundary values on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733024.png" /> can be expressed. Clearly, this apparatus will be different for different classes of analytic functions.
+
$  \epsilon > 0 $,
 +
of width  $  \pi - 2 \epsilon < \pi $,  
 +
with apex at the point  $  e ^ {i \theta } $,
 +
bisected by the radius drawn through the point  $  e ^ {i \theta } $.  
 +
Fatou's theorem cannot, in a certain sense, be improved upon; it was in fact shown by N.N. Luzin (1919) that for any set  $  E \subset  \Gamma $
 +
of measure zero on  $  \Gamma $
 +
there exists a bounded analytic function  $  f(z) $
 +
not having radial limits on  $  E $.
  
3) The uniqueness problem, or the problem of the properties that a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733025.png" /> should have such that two analytic functions of a given class coincide in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733026.png" /> if their boundary values on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733027.png" /> are identical.
+
The class of bounded analytic functions in a domain  $  D $
 +
is denoted by  $  B(D) $
 +
or  $  H  ^  \infty  (D) $.
 +
Following the results of Fatou, the next problem appeared to be the generalization of his theorems to wider classes of functions. One distinguishes between the following basic classes of functions in the unit disc  $  D $,
 +
which are related by proper inclusions:
  
The first result in the solution of the existence problem is the theorem of Fatou (1906): If an analytic function is bounded in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733029.png" />, then radial boundary, or limit, values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733030.png" /> exist almost everywhere with respect to the Lebesgue measure on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733031.png" />. It can be shown that, under these conditions, not only radial, but also angular boundary values, or boundary values along all non-tangential paths, exist almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733032.png" />. This means that, for almost-all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733034.png" /> tends to a definite limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733035.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733036.png" /> tends to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733037.png" /> while remaining within an arbitrary fixed angle
+
$$ \tag{1 }
 +
A (D) \subset  B (D)  = H  ^  \infty  (D)  \subset  \
 +
H  ^ {p} (D)  \subset  N  ^ {*} (D)  \subset  N (D).
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733038.png" /></td> </tr></table>
+
The class $  A(D) $
 +
is the class of single-valued analytic functions in  $  D $
 +
that are continuous in the closed domain  $  D \cup \Gamma = \overline{D}\; $.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733039.png" />, of width <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733040.png" />, with apex at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733041.png" />, bisected by the radius drawn through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733042.png" />. Fatou's theorem cannot, in a certain sense, be improved upon; it was in fact shown by N.N. Luzin (1919) that for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733043.png" /> of measure zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733044.png" /> there exists a bounded analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733045.png" /> not having radial limits on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733046.png" />.
+
The classes  $  H  ^ {p} (D) $,  
 +
for all positive numbers  $  p $,  
 +
are defined by the condition
  
The class of bounded analytic functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733047.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733048.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733049.png" />. Following the results of Fatou, the next problem appeared to be the generalization of his theorems to wider classes of functions. One distinguishes between the following basic classes of functions in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733050.png" />, which are related by proper inclusions:
+
$$ \tag{2 }
 +
\| f \| _ {p}  = \
 +
\sup _ {0 < r < 1 } \
 +
\left \{
 +
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
| f (re ^ {i \phi } ) |  ^ {p} \
 +
d \phi  \right \}  ^ {1/p}  = \
 +
C (f, p) <
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$
 +
< \
 +
+ \infty .
 +
$$
  
The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733052.png" /> is the class of single-valued analytic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733053.png" /> that are continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733054.png" />.
+
The proper inclusions  $  H  ^  \infty  \subset  H ^ {p _ {1} } \subset  H ^ {p _ {2} } $
 +
are valid for any  $  0 < p _ {1} < p _ {2} < + \infty $.  
 +
The classes  $  H  ^ {p} $
 +
were first introduced by G.H. Hardy (1915), and are often named Hardy classes. If  $  1 \leq  p < \infty $,
 +
one can introduce the norm (2) on  $  H  ^ {p} $,
 +
and the norm
  
The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733055.png" />, for all positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733056.png" />, are defined by the condition
+
$$
 +
\| f \| _  \infty  = \
 +
\| f \| _ {B}  = \
 +
\sup _ {z \in D } \
 +
| f(z) |
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
on  $  H  ^  \infty  $;  
 +
the classes  $  H  ^ {p} $,
 +
$  1 \leq  p \leq  + \infty $,
 +
have a natural structure of a vector space, and become Banach Hardy spaces. If  $  0 < p < 1 $,
 +
it is only possible to introduce the metric  $  \rho _ {p} (f, g) = \| f - g \| _ {p}  ^ {p} $
 +
on  $  H  ^ {p} $,
 +
which converts the latter into a complete metric non-normable space. The class of bounded analytic functions  $  B = H  ^  \infty  $
 +
is contained in any class  $  H  ^ {p} $,
 +
$  p > 0 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733058.png" /></td> </tr></table>
+
The class $  N(D) $
 +
of certain meromorphic functions  $  f(z) $
 +
in the unit disc  $  D $
 +
is said to be the class of functions of bounded characteristic; it was introduced in 1924 by R. Nevanlinna. The class  $  N(D) $
 +
can be characterized as the set of meromorphic functions  $  f(z) $
 +
in  $  D $
 +
that can be represented as the ratio of two bounded regular functions  $  f _ {1} (z) $
 +
and  $  f _ {2} (z) $
 +
in  $  D $,
 +
$  f(z) = f _ {1} (z)/f _ {2} (z) $.
  
The proper inclusions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733059.png" /> are valid for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733060.png" />. The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733061.png" /> were first introduced by G.H. Hardy (1915), and are often named Hardy classes. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733064.png" />, one can introduce the norm (2) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733065.png" />, and the norm
+
All regular functions  $  f(z) \in N(D) $
 +
form a subclass  $  N  ^ {*} (D) $,  
 +
and $  f(z) \in N  ^ {*} (D) $
 +
if and only if the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733066.png" /></td> </tr></table>
+
$$ \tag{3 }
 +
\sup _ {0 < r < 1 } \
  
on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733067.png" />; the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733069.png" />, have a natural structure of a vector space, and become Banach Hardy spaces. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733070.png" />, it is only possible to introduce the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733071.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733072.png" />, which converts the latter into a complete metric non-normable space. The class of bounded analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733073.png" /> is contained in any class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733075.png" />.
+
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
\mathop{\rm ln}  ^ {+}  | f (re ^ {i \phi } ) | \
 +
d \phi  = C (f)  < + \infty ,
 +
$$
  
The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733076.png" /> of certain meromorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733077.png" /> in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733078.png" /> is said to be the class of functions of bounded characteristic; it was introduced in 1924 by R. Nevanlinna. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733079.png" /> can be characterized as the set of meromorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733080.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733081.png" /> that can be represented as the ratio of two bounded regular functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733083.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733085.png" />.
+
where  $  \mathop{\rm ln}  ^ {+}  x = \mathop{\rm ln}  x $
 +
if  $  \mathop{\rm ln}  x \geq  0 $
 +
and  $  \mathop{\rm ln}  ^ {+}  x = 0 $
 +
if  $  \mathop{\rm ln}  x < 0 $,
 +
is fulfilled. The class $  N  ^ {*} (D) $
 +
contains all classes  $  H  ^ {p} $,
 +
0 < p \leq  + \infty $.
  
All regular functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733086.png" /> form a subclass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733087.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733088.png" /> if and only if the condition
+
The classes  $  H  ^ {p} $
 +
have the following generalization. Let  $  \psi (t) $
 +
be a strictly-convex function for  $  - \infty < t < + \infty $,
 +
i.e. a non-negative, convex, non-decreasing function such that  $  \psi (t) / t \rightarrow + \infty $
 +
as  $  t \rightarrow + \infty $.  
 +
The class  $  H _  \psi  (D) $
 +
is then defined by the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733089.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{2'}
 +
\sup _ {0 < r < 1 } \
 +
\psi  ^ {-1} \left \{
 +
\frac{1}{2 \pi }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733090.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733092.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733093.png" />, is fulfilled. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733094.png" /> contains all classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733096.png" />.
+
\int\limits _ { 0 } ^ { {2 }  \pi } \psi
 +
(  \mathop{\rm ln}  | f (re ^ {i \phi } ) | )  d \phi \right \}  = \
 +
C (f, \psi ) <
 +
$$
  
The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733097.png" /> have the following generalization. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733098.png" /> be a strictly-convex function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b01733099.png" />, i.e. a non-negative, convex, non-decreasing function such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330100.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330101.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330102.png" /> is then defined by the condition
+
$$
 +
< \
 +
+ \infty ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330103.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2prm)</td></tr></table>
+
compare with condition (2), where  $  \psi (t) = e  ^ {pt} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330104.png" /></td> </tr></table>
+
The principal result obtained for the problem of the existence of boundary values in the case of the unit disc  $  D $
 +
states that each meromorphic function  $  f (z) $
 +
of bounded characteristic in  $  D $
 +
has angular boundary values  $  f (e ^ {i \theta } ) $
 +
almost everywhere on  $  \Gamma $;  
 +
these boundary values are such that the function  $  \mathop{\rm ln}  | f ( e ^ {i \theta } ) | $
 +
is Lebesgue integrable on  $  \Gamma $.
 +
The following additional property is displayed by the classes  $  H  ^ {p} $,
 +
$  0 < p < + \infty $,
 +
or  $  H _  \psi  $:  
 +
The function  $  | f (e ^ {i \theta } ) |  ^ {p} $
 +
or, correspondingly,  $  \psi (  \mathop{\rm ln}  | f ( e ^ {i \theta } ) | )  ^ {p} $
 +
is Lebesgue integrable on  $  \Gamma $.
 +
For bounded functions  $  f(z) $,
 +
$  | f(z) | \leq  M $,
 +
one has instead of the above,  $  \mathop{\rm esssup}  | f (e ^ {i \theta } ) | \leq  M $,
 +
0 \leq  \theta \leq  2 \pi $.  
 +
Thus, condition (3) is the widest sufficient condition on the average increase of an analytic function  $  f(z) $,
 +
as  $  | z | \rightarrow 1 $,
 +
that ensures the existence of angular boundary values almost everywhere on  $  \Gamma $.
  
compare with condition (2), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330105.png" />.
+
It has been shown that condition (3) cannot be substantially weakened. Thus, it was shown by A. Zygmund that for an arbitrary increasing function  $  \psi (t) $,
 +
$  \psi (t) / t \rightarrow 0 $
 +
as  $  0 < t \uparrow + \infty $,
 +
there exists an analytic function  $  f(z) $
 +
in  $  D $
 +
such that
  
The principal result obtained for the problem of the existence of boundary values in the case of the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330106.png" /> states that each meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330107.png" /> of bounded characteristic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330108.png" /> has angular boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330109.png" /> almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330110.png" />; these boundary values are such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330111.png" /> is Lebesgue integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330112.png" />. The following additional property is displayed by the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330114.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330115.png" />: The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330116.png" /> or, correspondingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330117.png" /> is Lebesgue integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330118.png" />. For bounded functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330119.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330120.png" />, one has instead of the above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330122.png" />. Thus, condition (3) is the widest sufficient condition on the average increase of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330123.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330124.png" />, that ensures the existence of angular boundary values almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330125.png" />.
+
$$
 +
\sup _ {0 < r < 1 } \
 +
\int\limits _ { 0 } ^ { {2 }  \pi } \psi
 +
(  \mathop{\rm ln}  ^ {+}  | f  (re ^ {i \phi } ) | ) \
 +
d \phi  < + \infty ,
 +
$$
  
It has been shown that condition (3) cannot be substantially weakened. Thus, it was shown by A. Zygmund that for an arbitrary increasing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330127.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330128.png" />, there exists an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330129.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330130.png" /> such that
+
but which has no boundary values anywhere on  $  \Gamma $.
 +
Even if the maximum  $  M (r;  f) = \max \{ {| f(z) | } : {| z | = r } \} $
 +
grows as slowly as one pleases, there still exist analytic functions without radial boundary values.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330131.png" /></td> </tr></table>
+
The boundary representation of functions  $  f(z) $
 +
of class $  N(D) $,
 +
characterizing the functions of this class, has the form
  
but which has no boundary values anywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330132.png" />. Even if the maximum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330133.png" /> grows as slowly as one pleases, there still exist analytic functions without radial boundary values.
+
$$ \tag{4 }
 +
f (z)  = z  ^ {m} e ^ {i \lambda }
  
The boundary representation of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330134.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330135.png" />, characterizing the functions of this class, has the form
+
\frac{B _ {1} (z;  a _  \mu  ) }{B _ {2} (z;  b _  \nu  ) }
 +
\times
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330136.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$
 +
\times
 +
\mathop{\rm exp} 
 +
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi }  \mathop{\rm ln}
 +
| f (e ^ {i \theta } ) |
 +
\frac{e ^ {i \theta }
 +
+ z }{e ^ {i \theta } - z }
 +
  d \theta \times
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330137.png" /></td> </tr></table>
+
$$
 +
\times
 +
\mathop{\rm exp} 
 +
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi }
 +
\frac{e ^ {i \theta } + z }{e ^ {i \theta } - z }
 +
  d \Phi ( \theta ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330138.png" /></td> </tr></table>
+
where  $  m $
 +
is an integer,  $  m = k $
 +
if the point  $  z = 0 $
 +
is a zero of multiplicity  $  k $
 +
and  $  m = - k $
 +
if  $  z = 0 $
 +
is a pole of multiplicity  $  k $;  
 +
$  \lambda $
 +
is a real number;
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330139.png" /> is an integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330140.png" /> if the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330141.png" /> is a zero of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330143.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330144.png" /> is a pole of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330145.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330146.png" /> is a real number;
+
$$ \tag{5 }
 +
B _ {1} (z;  a _  \mu  )  = \
 +
\prod _ {\mu = 1 } ^  \infty 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330147.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
\frac{| a _  \mu  | }{a _  \mu  }
  
is the [[Blaschke product|Blaschke product]] taken over all the zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330148.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330149.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330150.png" /> taking into account their multiplicity; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330151.png" /> is the Blaschke product of type (5) taken over all poles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330152.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330153.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330154.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330155.png" /> is a singular function of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330156.png" /> with a derivative that vanishes almost everywhere. In (4), the last integral is of Lebesgue–Stieltjes type, while the first is of Lebesgue type.
+
\frac{a _  \mu  - z }{1 - \overline{ {a _  \mu  }}\; z }
  
It was shown by M.M. Dzhrbashyan [[#References|[10]]] that the theory of meromorphic functions of bounded characteristic can be considerably extended. It is possible, in fact, to introduce a family of classes of meromorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330157.png" /> depending on a continuous parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330159.png" />, and the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330160.png" /> are characterized by representations yielding (4) when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330161.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330162.png" /> increases, the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330163.png" /> become larger, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330164.png" /> becomes identical with Nevanlinna's class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330165.png" />.
+
$$
  
For analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330166.png" /> in the representation (4) one must put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330167.png" />. For the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330168.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330169.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330170.png" />, in the representation (4) one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330171.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330172.png" /> is a non-increasing function of the indicated type. See also [[Cauchy integral|Cauchy integral]].
+
is the [[Blaschke product|Blaschke product]] taken over all the zeros  $  a _  \mu  \neq 0 $
 +
of  $  f (z) $
 +
inside  $  D $
 +
taking into account their multiplicity;  $  B _ {2} (z;  b _  \nu  ) $
 +
is the Blaschke product of type (5) taken over all poles  $  b _  \nu  \neq 0 $
 +
of  $  f(z) $
 +
in  $  D $;
 +
and  $  \Phi ( \theta ) $
 +
is a singular function of bounded variation on  $  [0, 2 \pi ] $
 +
with a derivative that vanishes almost everywhere. In (4), the last integral is of Lebesgue–Stieltjes type, while the first is of Lebesgue type.
  
The first results in the uniqueness problem were obtained in 1916 by the brothers F. and M. Riesz: If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330173.png" /> has radial boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330174.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330175.png" /> of positive Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330176.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330177.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330178.png" />. The representation (4) makes it possible to extend this theorem to meromorphic functions of bounded characteristic (cf. [[Function of bounded form|Function of bounded form]]). N.N. Luzin (1919) constructed, for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330179.png" /> of measure zero, an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330180.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330181.png" /> everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330182.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330183.png" /> in an arbitrary manner, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330184.png" /> is not identically equal to zero. The deepest and most general boundary uniqueness theorems for meromorphic functions of general form were obtained in 1925 by Luzin and I.I. Privalov (cf. [[Uniqueness properties of analytic functions|Uniqueness properties of analytic functions]]; [[Luzin–Privalov theorems|Luzin–Privalov theorems]]).
+
It was shown by M.M. Dzhrbashyan [[#References|[10]]] that the theory of meromorphic functions of bounded characteristic can be considerably extended. It is possible, in fact, to introduce a family of classes of meromorphic functions  $  N _  \alpha  $
 +
depending on a continuous parameter  $  \alpha $,
 +
$  -1 < \alpha < + \infty $,
 +
and the classes  $  N _  \alpha  $
 +
are characterized by representations yielding (4) when  $  \alpha = 0 $.  
 +
As  $  \alpha $
 +
increases, the classes  $  N _  \alpha  $
 +
become larger, and  $  N _ {0} $
 +
becomes identical with Nevanlinna's class  $  N $.
  
Consider the case of an arbitrary plane domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330185.png" />; for the sake of brevity, only simply-connected domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330186.png" /> with a rectifiable boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330187.png" /> will be discussed. The conditions (2), (3) and (2prm) are equivalent to stipulating that the subharmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330188.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330189.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330190.png" />, respectively, have a [[Harmonic majorant|harmonic majorant]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330191.png" />. In such a form these conditions are fully suitable, and furnish a natural definition of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330192.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330193.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330194.png" /> in arbitrary domains. It is known that a rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330195.png" /> has a definite tangent and normal at almost all of its points. The inclusions (1) remain valid, as does Fatou's theorem on the almost-everywhere existence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330196.png" /> of angular boundary values for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330197.png" />. Here, as the bisectrix of the angular domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330198.png" /> the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330199.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330200.png" /> is taken. The Riesz uniqueness theorem for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330201.png" /> is also applicable.
+
For analytic functions  $  f(z) \in N  ^ {*} (D) $
 +
in the representation (4) one must put  $  B _ {2} (z;  b _  \nu  ) \equiv 1 $.
 +
For the functions $  f(z) \in H  ^ {p} $,  
 +
0 < p \leq  + \infty $,  
 +
or  $  \in H _  \psi  $,  
 +
in the representation (4) one has  $  B _ {2} (z;  b _  \nu  ) \equiv 1 $,  
 +
and $  \Phi ( \theta ) $
 +
is a non-increasing function of the indicated type. See also [[Cauchy integral|Cauchy integral]].
  
V.I. Smirnov also introduced the frequently employed classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330203.png" />, for the case of an arbitrary domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330204.png" />. These classes have the following definition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330205.png" /> if there exist a sequence of contours <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330207.png" />, such that
+
The first results in the uniqueness problem were obtained in 1916 by the brothers F. and M. Riesz: If a function  $  f(z) \in H  ^  \infty  $
 +
has radial boundary values  $  f (e ^ {i \theta } ) = 0 $
 +
on a set  $  E \subset  \Gamma $
 +
of positive Lebesgue measure on  $  \Gamma $,
 +
then  $  f(z) \equiv 0 $
 +
in  $  D $.  
 +
The representation (4) makes it possible to extend this theorem to meromorphic functions of bounded characteristic (cf. [[Function of bounded form|Function of bounded form]]). N.N. Luzin (1919) constructed, for any set  $  E \subset  \Gamma $
 +
of measure zero, an analytic function  $  f(z) $
 +
such that  $  f ( e ^ {i \theta } ) = 0 $
 +
everywhere on  $  E $
 +
as  $  z \rightarrow e ^ {i \theta } $
 +
in an arbitrary manner, but  $  f (z) $
 +
is not identically equal to zero. The deepest and most general boundary uniqueness theorems for meromorphic functions of general form were obtained in 1925 by Luzin and I.I. Privalov (cf. [[Uniqueness properties of analytic functions|Uniqueness properties of analytic functions]]; [[Luzin–Privalov theorems|Luzin–Privalov theorems]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330208.png" /></td> </tr></table>
+
Consider the case of an arbitrary plane domain  $  D $;  
 +
for the sake of brevity, only simply-connected domains  $  D $
 +
with a rectifiable boundary  $  \Gamma $
 +
will be discussed. The conditions (2), (3) and (2'}) are equivalent to stipulating that the subharmonic functions  $  {| f(z) | }  ^ {p} $,
 +
$  \mathop{\rm ln}  ^ {+}  | f(z) | $
 +
and  $  \psi (  \mathop{\rm ln}  | f (z) | ) $,
 +
respectively, have a [[Harmonic majorant|harmonic majorant]] in  $  D $.
 +
In such a form these conditions are fully suitable, and furnish a natural definition of the classes  $  H  ^ {p} $,
 +
$  N  ^ {*} $
 +
and  $  H _  \psi  $
 +
in arbitrary domains. It is known that a rectifiable curve  $  \Gamma $
 +
has a definite tangent and normal at almost all of its points. The inclusions (1) remain valid, as does Fatou's theorem on the almost-everywhere existence on  $  \Gamma $
 +
of angular boundary values for the class  $  N  ^ {*} $.  
 +
Here, as the bisectrix of the angular domains  $  \Delta ( \zeta , \epsilon ) $
 +
the normal to  $  \Gamma $
 +
at the point  $  \zeta \in \Gamma $
 +
is taken. The Riesz uniqueness theorem for the class  $  N  ^ {*} $
 +
is also applicable.
  
The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330209.png" /> are especially convenient in the study of problems of representation of functions in the form of a Cauchy integral.
+
V.I. Smirnov also introduced the frequently employed classes  $  E  ^ {p} $,
 +
$  p > 0 $,
 +
for the case of an arbitrary domain  $  D $.
 +
These classes have the following definition:  $  f(z) \in E  ^ {p} (D) $
 +
if there exist a sequence of contours  $  \{ \Gamma _ {j} \} \subset  D $,
 +
$  \Gamma _ {j} \rightarrow \Gamma $,
 +
such that
  
Of major interest is the study of the boundary properties of analytic functions realizing a conformal mapping. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330210.png" /> realize a conformal mapping of the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330211.png" /> onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330212.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330213.png" />-plane with a rectifiable boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330214.png" />. It has been shown, for example, that in such a case the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330215.png" /> belongs to the Hardy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330216.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330217.png" />, so that it is representable in the form (4) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330218.png" /> and a non-increasing singular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330219.png" />. Smirnov pointed out the importance of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330220.png" /> of such domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330221.png" /> for which this singular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330222.png" />. In 1937 M.V. Keldysh and M.A. Lavrent'ev constructed an example of a domain with a rectifiable boundary that is not included in the Smirnov classes just mentioned; this renders the characterization of domains of Smirnov type even more important.
+
$$
 +
\sup _ {\{ \Gamma _ {j} \} } \
 +
\int\limits _ {\Gamma _ {j} }
 +
| f (z) |  ^ {p}
 +
| dz |  = C (f, p) < + \infty .
 +
$$
  
Numerous workers also attempt to study boundary properties of analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330223.png" /> in several variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330224.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330225.png" /> be the unit [[Polydisc|polydisc]], and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330226.png" /> be its skeleton. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330227.png" /> of analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330228.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330229.png" /> is defined by the condition
+
The classes  $  E  ^ {p} $
 +
are especially convenient in the study of problems of representation of functions in the form of a Cauchy integral.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330230.png" /></td> </tr></table>
+
Of major interest is the study of the boundary properties of analytic functions realizing a conformal mapping. Let a function  $  z = F (w) $
 +
realize a conformal mapping of the unit disc  $  | w | < 1 $
 +
onto a domain  $  D $
 +
of the  $  z $-
 +
plane with a rectifiable boundary  $  \Gamma $.
 +
It has been shown, for example, that in such a case the derivative  $  F ^ { \prime } (w) $
 +
belongs to the Hardy class $  H  ^ {1} $
 +
in the disc  $  | w | < 1 $,
 +
so that it is representable in the form (4) with  $  B _ {2} \equiv 1 $
 +
and a non-increasing singular function  $  \Phi ( \theta ) $.
 +
Smirnov pointed out the importance of the class  $  S $
 +
of such domains  $  D $
 +
for which this singular function  $  \Phi ( \theta ) \equiv 0 $.
 +
In 1937 M.V. Keldysh and M.A. Lavrent'ev constructed an example of a domain with a rectifiable boundary that is not included in the Smirnov classes just mentioned; this renders the characterization of domains of Smirnov type even more important.
  
which is analogous to (3), while the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330231.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330232.png" /> by the condition of the type (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330233.png" /> for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330234.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330235.png" />):
+
Numerous workers also attempt to study boundary properties of analytic functions  $  f(z) $
 +
in several variables  $  z = (z _ {1} \dots z _ {n} ) $.
 +
Let  $  D = U  ^ {n} = \{ {z \in \mathbf C  ^ {n} } : {| z _ {j} | < 1,  j = 1 \dots n } \} $
 +
be the unit [[Polydisc|polydisc]], and let  $  T  ^ {n} = \{ {z \in \mathbf C  ^ {n} } : {| z _ {j} | = 1, j = 1 \dots n } \} $
 +
be its skeleton. The class  $  N  ^ {*} (D) $
 +
of analytic functions  $  f(z) $
 +
in  $  D $
 +
is defined by the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330236.png" /></td> </tr></table>
+
$$
 +
\sup _ {0 < r < 1 } \
 +
\int\limits _ {T  ^ {n} }
 +
\mathop{\rm ln}  ^ {+}  | f(rz) | \
 +
dm _ {n} (z)  = \
 +
C (f)  < + \infty ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330237.png" /> is the normalized [[Haar measure|Haar measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330238.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330239.png" />. Inclusions of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330240.png" /> are preserved. Analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330241.png" /> have "radial" boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330242.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330243.png" />, almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330244.png" /> with respect to the Haar measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330245.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330246.png" /> is summable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330247.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330248.png" />. Sufficiently simple and general characteristics for boundary representations and uniqueness properties of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330249.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330250.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330251.png" /> have not yet (1986) been found.
+
which is analogous to (3), while the classes  $  H  ^ {p} (D) $
 +
or  $  H _  \psi  (D) $
 +
by the condition of the type ( $  \psi (t) = e  ^ {pt} $
 +
for the case  $ H ^ {p} (D) $,  
 +
$  p > 0 $):
  
Many boundary properties may be applied to various generalizations of analytic functions, in particular to abstract analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330252.png" />, which have values in, say, a separable locally convex topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330253.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330254.png" />.
+
$$
 +
\sup _ {0 < r < 1 } \
 +
\psi  ^ {-1}
 +
\left \{ \int\limits _ {T  ^ {n} }
 +
\psi (  \mathop{\rm ln}  | f  (rz) |) \
 +
dm _ {n} (z) \right \}  = \
 +
C(f, \psi )  <  + \infty ,
 +
$$
 +
 
 +
where  $  m _ {n} $
 +
is the normalized [[Haar measure|Haar measure]] on  $  T  ^ {n} $,
 +
$  m _ {n} (T  ^ {n} ) = 1 $.
 +
Inclusions of the type  $  B = H  ^  \infty  \subset  H _  \psi  \subset  N  ^ {*} $
 +
are preserved. Analytic functions  $  f(z) \in N  ^ {*} (D) $
 +
have  "radial"  boundary values  $  f  ^ {*} (z) = \lim\limits _ {r \rightarrow 1 - 0 }  f(rz) $,
 +
$  z \in T  ^ {n} $,
 +
almost everywhere on  $  T  ^ {n} $
 +
with respect to the Haar measure  $  m _ {n} $;
 +
and  $  \mathop{\rm ln}  ^ {+}  | f  ^ {*} (z) | $
 +
is summable on  $  T _ {n} $
 +
with respect to  $  m _ {n} $.
 +
Sufficiently simple and general characteristics for boundary representations and uniqueness properties of functions  $  f(z) $
 +
in  $  U  ^ {n} $
 +
for  $  n > 1 $
 +
have not yet (1986) been found.
 +
 
 +
Many boundary properties may be applied to various generalizations of analytic functions, in particular to abstract analytic functions $  f: D \rightarrow X $,  
 +
which have values in, say, a separable locally convex topological space $  X $
 +
over the field $  \mathbf C $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Golubev,  "Univalent analytic functions. Automorphic functions" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.Ya. Khavinson,  "Analytic functions of bounded type"  ''Itogi Nauk. Mat. Anal. 1963''  (1965)  pp. 5–80  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. Nevanilinna,  "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Noshiro,  "Cluster sets" , Springer  (1960)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  G.R. MacLane,  "Asymptotic values of holomorphic functions" , ''Rice Univ. Studies, Math. Monographs'' , '''49''' :  1 , Rice Univ. , Houston  (1963)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.M. Dzhrbashyan,  "Integral transforms and representation of functions in the complex domain" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  W. Rudin,  "Function theory in polydiscs" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  G.M. Khenkin,  E.M. Chirka,  "Boundary properties of holomorphic functions of several complex variables"  ''J. Soviet Math.'' , '''5''' :  5  (1971)  pp. 612–687  ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''4'''  (1975)  pp. 13–142</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Golubev,  "Univalent analytic functions. Automorphic functions" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.Ya. Khavinson,  "Analytic functions of bounded type"  ''Itogi Nauk. Mat. Anal. 1963''  (1965)  pp. 5–80  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. Nevanilinna,  "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Noshiro,  "Cluster sets" , Springer  (1960)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  G.R. MacLane,  "Asymptotic values of holomorphic functions" , ''Rice Univ. Studies, Math. Monographs'' , '''49''' :  1 , Rice Univ. , Houston  (1963)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.M. Dzhrbashyan,  "Integral transforms and representation of functions in the complex domain" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  W. Rudin,  "Function theory in polydiscs" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  G.M. Khenkin,  E.M. Chirka,  "Boundary properties of holomorphic functions of several complex variables"  ''J. Soviet Math.'' , '''5''' :  5  (1971)  pp. 612–687  ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''4'''  (1975)  pp. 13–142</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Sokhotskii's theorem is better known as the Casorati–Weierstrass theorem. In (4) the first integral on the right-hand side is called the outer factor, while the other factors on the right-hand side form together the inner factor; the latter has absolute boundary values 1 almost-everywhere. If the outer factor in (4) equals 1, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330255.png" /> is called an inner function. Recent general references are [[#References|[a2]]], [[#References|[a3]]].
+
Sokhotskii's theorem is better known as the Casorati–Weierstrass theorem. In (4) the first integral on the right-hand side is called the outer factor, while the other factors on the right-hand side form together the inner factor; the latter has absolute boundary values 1 almost-everywhere. If the outer factor in (4) equals 1, $  f $
 +
is called an inner function. Recent general references are [[#References|[a2]]], [[#References|[a3]]].
  
For smooth domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330256.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330257.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330258.png" />, there is a strong version of Fatou's theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330259.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330260.png" />: Boundary values exist almost-everywhere (or even better than almost-everywhere) if the boundary is approached over admissible approach regions, see [[#References|[a5]]], [[#References|[a7]]]. In contrast with the one-dimensional case, zero sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330261.png" />-functions can be essentially different for different values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330262.png" />, see [[#References|[11]]], [[#References|[a5]]]. A characterization of zero sets of functions of Nevanlinna class on smooth strongly pseudo-convex domains was obtained by G.M. Khenkin and, independently, by H. Skoda, see [[#References|[a5]]]. A.B. Aleksandrov and E. Løw proved independently the existence of non-constant inner functions for the unit-ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330263.png" />, see [[#References|[a1]]], [[#References|[a4]]]. In an other direction, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330264.png" />-spaces have been introduced for the upper half-space in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330265.png" />, [[#References|[a6]]].
+
For smooth domains $  \Omega $
 +
in $  \mathbf C  ^ {n} $,  
 +
$  n \geq  2 $,  
 +
there is a strong version of Fatou's theorem for $  H  ^ {p} ( \Omega ) $
 +
and $  N  ^ {*} ( \Omega ) $:  
 +
Boundary values exist almost-everywhere (or even better than almost-everywhere) if the boundary is approached over admissible approach regions, see [[#References|[a5]]], [[#References|[a7]]]. In contrast with the one-dimensional case, zero sets of $  H  ^ {p} $-
 +
functions can be essentially different for different values of $  p $,  
 +
see [[#References|[11]]], [[#References|[a5]]]. A characterization of zero sets of functions of Nevanlinna class on smooth strongly pseudo-convex domains was obtained by G.M. Khenkin and, independently, by H. Skoda, see [[#References|[a5]]]. A.B. Aleksandrov and E. Løw proved independently the existence of non-constant inner functions for the unit-ball in $  \mathbf C  ^ {n} $,  
 +
see [[#References|[a1]]], [[#References|[a4]]]. In an other direction, $  H  ^ {p} $-
 +
spaces have been introduced for the upper half-space in $  \mathbf R  ^ {n} $,  
 +
[[#References|[a6]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.B. Aleksandrov,  "The existence of inner functions in the ball"  ''Math. USSR Sb.'' , '''46'''  (1983)  pp. 143–159</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.-B. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Koosis,  "Introduction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330266.png" />-spaces. With an appendix on Wolff's proof of the corona theorem" , Cambridge Univ. Press  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Løw,  "A construction of inner functions on the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330267.png" />"  ''Invent. Math.'' , '''67'''  (1982)  pp. 223–229</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Rudin,  "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330268.png" />" , Springer  (1980)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E.M. Stein,  "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)  pp. Chapt. 7</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E.M. Stein,  "Boundary behavior of holomorphic functions of several complex variables" , Princeton Univ. Press  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.B. Aleksandrov,  "The existence of inner functions in the ball"  ''Math. USSR Sb.'' , '''46'''  (1983)  pp. 143–159</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.-B. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Koosis,  "Introduction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330266.png" />-spaces. With an appendix on Wolff's proof of the corona theorem" , Cambridge Univ. Press  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Løw,  "A construction of inner functions on the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330267.png" />"  ''Invent. Math.'' , '''67'''  (1982)  pp. 223–229</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Rudin,  "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017330/b017330268.png" />" , Springer  (1980)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E.M. Stein,  "Singular integrals and differentiability properties of functions" , Princeton Univ. Press  (1970)  pp. Chapt. 7</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E.M. Stein,  "Boundary behavior of holomorphic functions of several complex variables" , Princeton Univ. Press  (1972)</TD></TR></table>

Latest revision as of 06:29, 30 May 2020


Properties of analytic functions that are displayed as the function approaches the boundary of its domain of definition.

It can be said that the study of boundary properties of analytic functions, understood in the widest sense of the word, began with the Sokhotskii theorem and the Picard theorem about the behaviour of analytic functions in a neighbourhood of isolated essential singular points (cf. Essential singular point), which were obtained in the second half of the 19th century. The terms relevant to this approach to the study of boundary properties of analytic functions — which is now called the theory of prime ends and cluster sets (cf. Limit elements) — first appeared in a course given by P. Painlevé in 1895. The dissertation of P. Fatou (1906) is the first systematic study of certain boundary properties of analytic functions in a neighbourhood of the continuous boundary of their domain of definition. The theory of boundary properties made considerable advances in the first third of the 20th century, owing to the work of several scientists; it resumed its rapid advance in the second half of that century, accompanied by the appearance of new ideas and methods, novel directions and objects of study. Its development is closely connected with various fields of mathematical analysis and mathematics in general, first and foremost with probability theory, the theory of harmonic functions, the theory of conformal mapping, boundary value problems of analytic function theory, potential theory, value-distribution theory, Riemann surfaces, subharmonic functions and function algebras. The theory of boundary properties of analytic functions is closely connected with various fields of application of mathematics by way of boundary value problems.

Since the study of boundary properties is connected, in the first place, with the geometry of the boundary $ \Gamma $ of the domain of definition $ D $ of an analytic function $ f(z) $ in one complex variable $ z $, three main approaches can be distinguished in the theory of boundary properties of analytic functions.

a) The study of the behaviour of $ f(z) $ in a neighbourhood of an isolated boundary point $ a \in \Gamma $. The most important case is that of an essential singular point $ a $, which is dealt with in the theorems of Sokhotskii, Picard, Julia, and Iversen (cf. Sokhotskii theorem; Picard theorem; Julia theorem; Iversen theorem).

b) The study of the behaviour of $ f(z) $ in the case when the boundary $ \Gamma $ is an everywhere-discontinuous set. V.V. Golubev's dissertation Single-valued analytic functions with a perfect set of singular points (1916, cf. [1]) was of great importance in this connection.

c) The study of the behaviour of $ f(z) $ when the domain $ D $ is bounded by a continuous closed curve $ \Gamma $ and, in particular, by the unit circle.

Cases a) and c) are, in a sense, extreme cases, while case b) is intermediate. Case c), which is discussed below, has been the subject of most intense study.

Let an analytic function $ f(z) $ be defined in a finite simply-connected domain $ D $ of the complex $ z $- plane bounded by a rectifiable Jordan curve $ \Gamma $. The following problems are fundamental in the classical approach to the study of boundary properties of analytic functions.

1) The problem of the existence of boundary values, i.e. the question under which conditions and in which sense the boundary values of $ f(z) $ exist as the point $ z $ approaches $ \Gamma $. This problem, as well as the succeeding ones, can be formulated in a different manner as the problem of identifying sufficiently extensive classes of analytic functions in $ D $ that have, in some sense, boundary values for sufficiently large sets of points on $ \Gamma $.

2) The problem of boundary representation of $ f(z) $, i.e. the problem under which conditions and with what kind of analytic apparatus the dependence of $ f(z) $ on its boundary values on $ \Gamma $ can be expressed. Clearly, this apparatus will be different for different classes of analytic functions.

3) The uniqueness problem, or the problem of the properties that a set $ E \subset \Gamma $ should have such that two analytic functions of a given class coincide in $ D $ if their boundary values on $ E $ are identical.

The first result in the solution of the existence problem is the theorem of Fatou (1906): If an analytic function is bounded in the unit disc $ D = \{ {z } : {| z | < 1 } \} $, $ | f(z) | \leq M $, then radial boundary, or limit, values $ f(e ^ {i \theta } ) = \lim\limits _ {r \rightarrow 1 - 0 } f (re ^ {i \theta } ) $ exist almost everywhere with respect to the Lebesgue measure on the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $. It can be shown that, under these conditions, not only radial, but also angular boundary values, or boundary values along all non-tangential paths, exist almost everywhere on $ \Gamma $. This means that, for almost-all points $ e ^ {i \theta } \in \Gamma $, $ f(z) $ tends to a definite limit $ f ( e ^ {i \theta } ) $ as $ z $ tends to the point $ e ^ {i \theta } $ while remaining within an arbitrary fixed angle

$$ \Delta (e ^ {i \theta } , \epsilon ) = \ \{ | z | < 1 \} \cap \left \{ | \mathop{\rm arg} (e ^ {i \theta } - z) | < { \frac \pi {2} } - \epsilon \right \} , $$

$ \epsilon > 0 $, of width $ \pi - 2 \epsilon < \pi $, with apex at the point $ e ^ {i \theta } $, bisected by the radius drawn through the point $ e ^ {i \theta } $. Fatou's theorem cannot, in a certain sense, be improved upon; it was in fact shown by N.N. Luzin (1919) that for any set $ E \subset \Gamma $ of measure zero on $ \Gamma $ there exists a bounded analytic function $ f(z) $ not having radial limits on $ E $.

The class of bounded analytic functions in a domain $ D $ is denoted by $ B(D) $ or $ H ^ \infty (D) $. Following the results of Fatou, the next problem appeared to be the generalization of his theorems to wider classes of functions. One distinguishes between the following basic classes of functions in the unit disc $ D $, which are related by proper inclusions:

$$ \tag{1 } A (D) \subset B (D) = H ^ \infty (D) \subset \ H ^ {p} (D) \subset N ^ {*} (D) \subset N (D). $$

The class $ A(D) $ is the class of single-valued analytic functions in $ D $ that are continuous in the closed domain $ D \cup \Gamma = \overline{D}\; $.

The classes $ H ^ {p} (D) $, for all positive numbers $ p $, are defined by the condition

$$ \tag{2 } \| f \| _ {p} = \ \sup _ {0 < r < 1 } \ \left \{ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } | f (re ^ {i \phi } ) | ^ {p} \ d \phi \right \} ^ {1/p} = \ C (f, p) < $$

$$ < \ + \infty . $$

The proper inclusions $ H ^ \infty \subset H ^ {p _ {1} } \subset H ^ {p _ {2} } $ are valid for any $ 0 < p _ {1} < p _ {2} < + \infty $. The classes $ H ^ {p} $ were first introduced by G.H. Hardy (1915), and are often named Hardy classes. If $ 1 \leq p < \infty $, one can introduce the norm (2) on $ H ^ {p} $, and the norm

$$ \| f \| _ \infty = \ \| f \| _ {B} = \ \sup _ {z \in D } \ | f(z) | $$

on $ H ^ \infty $; the classes $ H ^ {p} $, $ 1 \leq p \leq + \infty $, have a natural structure of a vector space, and become Banach Hardy spaces. If $ 0 < p < 1 $, it is only possible to introduce the metric $ \rho _ {p} (f, g) = \| f - g \| _ {p} ^ {p} $ on $ H ^ {p} $, which converts the latter into a complete metric non-normable space. The class of bounded analytic functions $ B = H ^ \infty $ is contained in any class $ H ^ {p} $, $ p > 0 $.

The class $ N(D) $ of certain meromorphic functions $ f(z) $ in the unit disc $ D $ is said to be the class of functions of bounded characteristic; it was introduced in 1924 by R. Nevanlinna. The class $ N(D) $ can be characterized as the set of meromorphic functions $ f(z) $ in $ D $ that can be represented as the ratio of two bounded regular functions $ f _ {1} (z) $ and $ f _ {2} (z) $ in $ D $, $ f(z) = f _ {1} (z)/f _ {2} (z) $.

All regular functions $ f(z) \in N(D) $ form a subclass $ N ^ {*} (D) $, and $ f(z) \in N ^ {*} (D) $ if and only if the condition

$$ \tag{3 } \sup _ {0 < r < 1 } \ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} ^ {+} | f (re ^ {i \phi } ) | \ d \phi = C (f) < + \infty , $$

where $ \mathop{\rm ln} ^ {+} x = \mathop{\rm ln} x $ if $ \mathop{\rm ln} x \geq 0 $ and $ \mathop{\rm ln} ^ {+} x = 0 $ if $ \mathop{\rm ln} x < 0 $, is fulfilled. The class $ N ^ {*} (D) $ contains all classes $ H ^ {p} $, $ 0 < p \leq + \infty $.

The classes $ H ^ {p} $ have the following generalization. Let $ \psi (t) $ be a strictly-convex function for $ - \infty < t < + \infty $, i.e. a non-negative, convex, non-decreasing function such that $ \psi (t) / t \rightarrow + \infty $ as $ t \rightarrow + \infty $. The class $ H _ \psi (D) $ is then defined by the condition

$$ \tag{2'} \sup _ {0 < r < 1 } \ \psi ^ {-1} \left \{ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } \psi ( \mathop{\rm ln} | f (re ^ {i \phi } ) | ) d \phi \right \} = \ C (f, \psi ) < $$

$$ < \ + \infty , $$

compare with condition (2), where $ \psi (t) = e ^ {pt} $.

The principal result obtained for the problem of the existence of boundary values in the case of the unit disc $ D $ states that each meromorphic function $ f (z) $ of bounded characteristic in $ D $ has angular boundary values $ f (e ^ {i \theta } ) $ almost everywhere on $ \Gamma $; these boundary values are such that the function $ \mathop{\rm ln} | f ( e ^ {i \theta } ) | $ is Lebesgue integrable on $ \Gamma $. The following additional property is displayed by the classes $ H ^ {p} $, $ 0 < p < + \infty $, or $ H _ \psi $: The function $ | f (e ^ {i \theta } ) | ^ {p} $ or, correspondingly, $ \psi ( \mathop{\rm ln} | f ( e ^ {i \theta } ) | ) ^ {p} $ is Lebesgue integrable on $ \Gamma $. For bounded functions $ f(z) $, $ | f(z) | \leq M $, one has instead of the above, $ \mathop{\rm esssup} | f (e ^ {i \theta } ) | \leq M $, $ 0 \leq \theta \leq 2 \pi $. Thus, condition (3) is the widest sufficient condition on the average increase of an analytic function $ f(z) $, as $ | z | \rightarrow 1 $, that ensures the existence of angular boundary values almost everywhere on $ \Gamma $.

It has been shown that condition (3) cannot be substantially weakened. Thus, it was shown by A. Zygmund that for an arbitrary increasing function $ \psi (t) $, $ \psi (t) / t \rightarrow 0 $ as $ 0 < t \uparrow + \infty $, there exists an analytic function $ f(z) $ in $ D $ such that

$$ \sup _ {0 < r < 1 } \ \int\limits _ { 0 } ^ { {2 } \pi } \psi ( \mathop{\rm ln} ^ {+} | f (re ^ {i \phi } ) | ) \ d \phi < + \infty , $$

but which has no boundary values anywhere on $ \Gamma $. Even if the maximum $ M (r; f) = \max \{ {| f(z) | } : {| z | = r } \} $ grows as slowly as one pleases, there still exist analytic functions without radial boundary values.

The boundary representation of functions $ f(z) $ of class $ N(D) $, characterizing the functions of this class, has the form

$$ \tag{4 } f (z) = z ^ {m} e ^ {i \lambda } \frac{B _ {1} (z; a _ \mu ) }{B _ {2} (z; b _ \nu ) } \times $$

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

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

where $ m $ is an integer, $ m = k $ if the point $ z = 0 $ is a zero of multiplicity $ k $ and $ m = - k $ if $ z = 0 $ is a pole of multiplicity $ k $; $ \lambda $ is a real number;

$$ \tag{5 } B _ {1} (z; a _ \mu ) = \ \prod _ {\mu = 1 } ^ \infty \frac{| a _ \mu | }{a _ \mu } \frac{a _ \mu - z }{1 - \overline{ {a _ \mu }}\; z } $$

is the Blaschke product taken over all the zeros $ a _ \mu \neq 0 $ of $ f (z) $ inside $ D $ taking into account their multiplicity; $ B _ {2} (z; b _ \nu ) $ is the Blaschke product of type (5) taken over all poles $ b _ \nu \neq 0 $ of $ f(z) $ in $ D $; and $ \Phi ( \theta ) $ is a singular function of bounded variation on $ [0, 2 \pi ] $ with a derivative that vanishes almost everywhere. In (4), the last integral is of Lebesgue–Stieltjes type, while the first is of Lebesgue type.

It was shown by M.M. Dzhrbashyan [10] that the theory of meromorphic functions of bounded characteristic can be considerably extended. It is possible, in fact, to introduce a family of classes of meromorphic functions $ N _ \alpha $ depending on a continuous parameter $ \alpha $, $ -1 < \alpha < + \infty $, and the classes $ N _ \alpha $ are characterized by representations yielding (4) when $ \alpha = 0 $. As $ \alpha $ increases, the classes $ N _ \alpha $ become larger, and $ N _ {0} $ becomes identical with Nevanlinna's class $ N $.

For analytic functions $ f(z) \in N ^ {*} (D) $ in the representation (4) one must put $ B _ {2} (z; b _ \nu ) \equiv 1 $. For the functions $ f(z) \in H ^ {p} $, $ 0 < p \leq + \infty $, or $ \in H _ \psi $, in the representation (4) one has $ B _ {2} (z; b _ \nu ) \equiv 1 $, and $ \Phi ( \theta ) $ is a non-increasing function of the indicated type. See also Cauchy integral.

The first results in the uniqueness problem were obtained in 1916 by the brothers F. and M. Riesz: If a function $ f(z) \in H ^ \infty $ has radial boundary values $ f (e ^ {i \theta } ) = 0 $ on a set $ E \subset \Gamma $ of positive Lebesgue measure on $ \Gamma $, then $ f(z) \equiv 0 $ in $ D $. The representation (4) makes it possible to extend this theorem to meromorphic functions of bounded characteristic (cf. Function of bounded form). N.N. Luzin (1919) constructed, for any set $ E \subset \Gamma $ of measure zero, an analytic function $ f(z) $ such that $ f ( e ^ {i \theta } ) = 0 $ everywhere on $ E $ as $ z \rightarrow e ^ {i \theta } $ in an arbitrary manner, but $ f (z) $ is not identically equal to zero. The deepest and most general boundary uniqueness theorems for meromorphic functions of general form were obtained in 1925 by Luzin and I.I. Privalov (cf. Uniqueness properties of analytic functions; Luzin–Privalov theorems).

Consider the case of an arbitrary plane domain $ D $; for the sake of brevity, only simply-connected domains $ D $ with a rectifiable boundary $ \Gamma $ will be discussed. The conditions (2), (3) and (2'}) are equivalent to stipulating that the subharmonic functions $ {| f(z) | } ^ {p} $, $ \mathop{\rm ln} ^ {+} | f(z) | $ and $ \psi ( \mathop{\rm ln} | f (z) | ) $, respectively, have a harmonic majorant in $ D $. In such a form these conditions are fully suitable, and furnish a natural definition of the classes $ H ^ {p} $, $ N ^ {*} $ and $ H _ \psi $ in arbitrary domains. It is known that a rectifiable curve $ \Gamma $ has a definite tangent and normal at almost all of its points. The inclusions (1) remain valid, as does Fatou's theorem on the almost-everywhere existence on $ \Gamma $ of angular boundary values for the class $ N ^ {*} $. Here, as the bisectrix of the angular domains $ \Delta ( \zeta , \epsilon ) $ the normal to $ \Gamma $ at the point $ \zeta \in \Gamma $ is taken. The Riesz uniqueness theorem for the class $ N ^ {*} $ is also applicable.

V.I. Smirnov also introduced the frequently employed classes $ E ^ {p} $, $ p > 0 $, for the case of an arbitrary domain $ D $. These classes have the following definition: $ f(z) \in E ^ {p} (D) $ if there exist a sequence of contours $ \{ \Gamma _ {j} \} \subset D $, $ \Gamma _ {j} \rightarrow \Gamma $, such that

$$ \sup _ {\{ \Gamma _ {j} \} } \ \int\limits _ {\Gamma _ {j} } | f (z) | ^ {p} | dz | = C (f, p) < + \infty . $$

The classes $ E ^ {p} $ are especially convenient in the study of problems of representation of functions in the form of a Cauchy integral.

Of major interest is the study of the boundary properties of analytic functions realizing a conformal mapping. Let a function $ z = F (w) $ realize a conformal mapping of the unit disc $ | w | < 1 $ onto a domain $ D $ of the $ z $- plane with a rectifiable boundary $ \Gamma $. It has been shown, for example, that in such a case the derivative $ F ^ { \prime } (w) $ belongs to the Hardy class $ H ^ {1} $ in the disc $ | w | < 1 $, so that it is representable in the form (4) with $ B _ {2} \equiv 1 $ and a non-increasing singular function $ \Phi ( \theta ) $. Smirnov pointed out the importance of the class $ S $ of such domains $ D $ for which this singular function $ \Phi ( \theta ) \equiv 0 $. In 1937 M.V. Keldysh and M.A. Lavrent'ev constructed an example of a domain with a rectifiable boundary that is not included in the Smirnov classes just mentioned; this renders the characterization of domains of Smirnov type even more important.

Numerous workers also attempt to study boundary properties of analytic functions $ f(z) $ in several variables $ z = (z _ {1} \dots z _ {n} ) $. Let $ D = U ^ {n} = \{ {z \in \mathbf C ^ {n} } : {| z _ {j} | < 1, j = 1 \dots n } \} $ be the unit polydisc, and let $ T ^ {n} = \{ {z \in \mathbf C ^ {n} } : {| z _ {j} | = 1, j = 1 \dots n } \} $ be its skeleton. The class $ N ^ {*} (D) $ of analytic functions $ f(z) $ in $ D $ is defined by the condition

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

which is analogous to (3), while the classes $ H ^ {p} (D) $ or $ H _ \psi (D) $ by the condition of the type ( $ \psi (t) = e ^ {pt} $ for the case $ H ^ {p} (D) $, $ p > 0 $):

$$ \sup _ {0 < r < 1 } \ \psi ^ {-1} \left \{ \int\limits _ {T ^ {n} } \psi ( \mathop{\rm ln} | f (rz) |) \ dm _ {n} (z) \right \} = \ C(f, \psi ) < + \infty , $$

where $ m _ {n} $ is the normalized Haar measure on $ T ^ {n} $, $ m _ {n} (T ^ {n} ) = 1 $. Inclusions of the type $ B = H ^ \infty \subset H _ \psi \subset N ^ {*} $ are preserved. Analytic functions $ f(z) \in N ^ {*} (D) $ have "radial" boundary values $ f ^ {*} (z) = \lim\limits _ {r \rightarrow 1 - 0 } f(rz) $, $ z \in T ^ {n} $, almost everywhere on $ T ^ {n} $ with respect to the Haar measure $ m _ {n} $; and $ \mathop{\rm ln} ^ {+} | f ^ {*} (z) | $ is summable on $ T _ {n} $ with respect to $ m _ {n} $. Sufficiently simple and general characteristics for boundary representations and uniqueness properties of functions $ f(z) $ in $ U ^ {n} $ for $ n > 1 $ have not yet (1986) been found.

Many boundary properties may be applied to various generalizations of analytic functions, in particular to abstract analytic functions $ f: D \rightarrow X $, which have values in, say, a separable locally convex topological space $ X $ over the field $ \mathbf C $.

References

[1] V.V. Golubev, "Univalent analytic functions. Automorphic functions" , Moscow (1961) (In Russian)
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[3] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[4] S.Ya. Khavinson, "Analytic functions of bounded type" Itogi Nauk. Mat. Anal. 1963 (1965) pp. 5–80 (In Russian)
[5] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[6] K. Noshiro, "Cluster sets" , Springer (1960)
[7] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9
[8] G.R. MacLane, "Asymptotic values of holomorphic functions" , Rice Univ. Studies, Math. Monographs , 49 : 1 , Rice Univ. , Houston (1963)
[9] A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)
[10] M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)
[11] W. Rudin, "Function theory in polydiscs" , Benjamin (1969)
[12] G.M. Khenkin, E.M. Chirka, "Boundary properties of holomorphic functions of several complex variables" J. Soviet Math. , 5 : 5 (1971) pp. 612–687 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 4 (1975) pp. 13–142

Comments

Sokhotskii's theorem is better known as the Casorati–Weierstrass theorem. In (4) the first integral on the right-hand side is called the outer factor, while the other factors on the right-hand side form together the inner factor; the latter has absolute boundary values 1 almost-everywhere. If the outer factor in (4) equals 1, $ f $ is called an inner function. Recent general references are [a2], [a3].

For smooth domains $ \Omega $ in $ \mathbf C ^ {n} $, $ n \geq 2 $, there is a strong version of Fatou's theorem for $ H ^ {p} ( \Omega ) $ and $ N ^ {*} ( \Omega ) $: Boundary values exist almost-everywhere (or even better than almost-everywhere) if the boundary is approached over admissible approach regions, see [a5], [a7]. In contrast with the one-dimensional case, zero sets of $ H ^ {p} $- functions can be essentially different for different values of $ p $, see [11], [a5]. A characterization of zero sets of functions of Nevanlinna class on smooth strongly pseudo-convex domains was obtained by G.M. Khenkin and, independently, by H. Skoda, see [a5]. A.B. Aleksandrov and E. Løw proved independently the existence of non-constant inner functions for the unit-ball in $ \mathbf C ^ {n} $, see [a1], [a4]. In an other direction, $ H ^ {p} $- spaces have been introduced for the upper half-space in $ \mathbf R ^ {n} $, [a6].

References

[a1] A.B. Aleksandrov, "The existence of inner functions in the ball" Math. USSR Sb. , 46 (1983) pp. 143–159
[a2] J.-B. Garnett, "Bounded analytic functions" , Acad. Press (1981)
[a3] P. Koosis, "Introduction to -spaces. With an appendix on Wolff's proof of the corona theorem" , Cambridge Univ. Press (1980)
[a4] E. Løw, "A construction of inner functions on the unit ball in " Invent. Math. , 67 (1982) pp. 223–229
[a5] W. Rudin, "Function theory in the unit ball in " , Springer (1980)
[a6] E.M. Stein, "Singular integrals and differentiability properties of functions" , Princeton Univ. Press (1970) pp. Chapt. 7
[a7] E.M. Stein, "Boundary behavior of holomorphic functions of several complex variables" , Princeton Univ. Press (1972)
How to Cite This Entry:
Boundary properties of analytic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_properties_of_analytic_functions&oldid=46130
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article