Namespaces
Variants
Actions

Difference between revisions of "Subordination principle"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
One of the forms of the [[Lindelöf principle|Lindelöf principle]], which employs the concept of subordination of functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s0909901.png" /> be any function regular in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s0909902.png" /> and satisfying the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s0909903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s0909904.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s0909905.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s0909906.png" /> be a meromorphic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s0909907.png" />. If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s0909908.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s0909909.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099010.png" /> is called subordinate to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099011.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099012.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099013.png" /> is called the subordinating function. This subordination relation is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099014.png" />. In the simplest case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099015.png" /> is a [[Univalent function|univalent function]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099016.png" />, this relation simply means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099017.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099018.png" /> does not take any values in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099019.png" /> that are not taken there by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099020.png" />. The following basic theorems apply.
+
<!--
 +
s0909901.png
 +
$#A+1 = 68 n = 1
 +
$#C+1 = 68 : ~/encyclopedia/old_files/data/S090/S.0900990 Subordination principle
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
One of the forms of the [[Lindelöf principle|Lindelöf principle]], which employs the concept of subordination of functions. Let $  \omega ( x) $
 +
be any function regular in the disc $  | z | < 1 $
 +
and satisfying the conditions $  \omega ( 0) = 0 $,  
 +
$  | \omega ( z) | < 1 $
 +
in $  | z | < 1 $;  
 +
let $  F( z) $
 +
be a meromorphic function in $  | z | < 1 $.  
 +
If the function $  f( z) $
 +
has the form $  f( z) = F( \omega ( z)) $,  
 +
then $  f( z) $
 +
is called subordinate to the function $  F( z) $
 +
in the disc $  | z | < 1 $,  
 +
while $  F( z) $
 +
is called the subordinating function. This subordination relation is denoted by $  f( z) \prec F( z) $.  
 +
In the simplest case where $  F( z) $
 +
is a [[Univalent function|univalent function]] in $  | z | < 1 $,  
 +
this relation simply means that $  f( 0) = F( 0) $
 +
and that $  f( z) $
 +
does not take any values in the disc $  | z | < 1 $
 +
that are not taken there by $  F( z) $.  
 +
The following basic theorems apply.
  
 
===Theorem 1.===
 
===Theorem 1.===
Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099021.png" /> be meromorphic in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099022.png" /> and map it on the [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099023.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099024.png" /> be the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099025.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099028.png" />, then the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099030.png" /> (under [[Analytic continuation|analytic continuation]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099031.png" />) lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099032.png" />, and the boundary points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099033.png" /> are obtained only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099035.png" /> [[#References|[2]]].
+
Let the function $  w = F( z) $
 +
be meromorphic in the disc $  | z | < 1 $
 +
and map it on the [[Riemann surface|Riemann surface]] $  G( F  ) $.  
 +
Let $  G _ {r} ( F  ) $
 +
be the part of $  G( F  ) $
 +
corresponding to $  | z | \leq  r $,  
 +
$  r < 1 $.  
 +
If $  f( z) \prec F( z) $,  
 +
then the values of $  f( z) $
 +
in $  | z | \leq  r $(
 +
under [[Analytic continuation|analytic continuation]] from $  f( 0) = F( 0) $)
 +
lie in $  G _ {r} ( F  ) $,  
 +
and the boundary points in $  G _ {r} ( F  ) $
 +
are obtained only for $  f( z) = F( \epsilon z) $,  
 +
$  | \epsilon | = 1 $[[#References|[2]]].
  
 
===Theorem 2.===
 
===Theorem 2.===
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099036.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099037.png" /> is regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099039.png" />, then setting
+
If $  f( z) \prec F( z) $
 +
and if $  F( z) $
 +
is regular in $  | z | \leq  r $,  
 +
$  r < 1 $,  
 +
then setting
  
<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/s/s090/s090990/s09099040.png" /></td> </tr></table>
+
$$
 +
M _  \lambda  ( r, f  )  = \
 +
\left \{
 +
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi } |
 +
f( re ^ {i \theta } ) |  ^  \lambda  d \phi \right \} ^ {1/ \lambda } ,\ \
 +
\lambda \geq  0 ,
 +
$$
  
one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099043.png" />, [[#References|[1]]].
+
one has $  M _  \lambda  ( r, f  ) \leq  M _  \lambda  ( r, F  ) $,
 +
$  \lambda \geq  0 $,  
 +
0 \leq  r < 1 $,  
 +
[[#References|[1]]].
  
 
===Theorem 3.===
 
===Theorem 3.===
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099045.png" /> is regular at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099046.png" />, then for the coefficients of the expansions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099048.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099050.png" /> [[#References|[2]]].
+
If $  f( z) \prec F( z) $
 
+
and $  F( z) $
The general theory of subordination is useful in considering the set of values taken or produced by an analytic function. The subordination relation can be used in two different ways. First, one can start from a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099051.png" /> and examine the behaviour of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099052.png" /> subordinate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099053.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099054.png" /> is completely known, then the region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099055.png" /> in which the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099056.png" /> lie is also known (Theorem 1) as well as an upper bound on the integral means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099057.png" /> (Theorem 2). If also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099058.png" /> is regular at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099059.png" />, there are upper bounds for the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099060.png" /> (Theorem 3). Secondly, one can consider a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099061.png" /> that is meromorphic in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099062.png" /> and whose properties imply that it cannot be subordinate to a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099063.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099064.png" />. If here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099065.png" />, for example, is univalent, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099066.png" /> necessarily takes values outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099067.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099068.png" />. These ideas of using the subordination relation illustrate the subordination principle and can be extended to multiply-connected domains [[#References|[3]]].
+
is regular at $  z = 0 $,  
 
+
then for the coefficients of the expansions $  f( z) = \sum _ {n=0} ^  \infty  a _ {n} z  ^ {n} $,  
====References====
+
$ F( z) = \sum _ {n=0} ^ \infty A _ {n} z ^ {n} $
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.E. Littlewood,  "On inequalities in the theory of functions"  ''Proc. Lond. Math. Soc. (2)'' , '''23'''  (1925)  pp. 481–519</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W. Rogosinski,  "On subordinate functions" ''Proc. Cambridge Philos. Soc.'' , '''35''' (1939)  pp. 1–26</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu. Alenitsyn,  "A generalization of the subordination principle to multiply-connected domains" ''Trudy Mat. Inst. Steklov.'' , '''60'''  (1961) pp. 5–21 (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Rogosinski,  ''Schr. K. Gelehrt. Gesellsch. Naturwiss. Kl.'' , '''8''' :  (1931)  pp. 1–31</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Rogosinski,  "On a theorem of Bieberbach–Eilenberg" ''J. Lond. Math. Soc.'' , '''14''' :  53 (1939)  pp. 4–11</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> W. Rogosinski,  "On the coefficients of subordinate functions" ''Proc. London Math. Soc.'' , '''48''' (1943)  pp. 48–82</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.E. Littlewood,  "Lectures on the theory of functions" , Oxford Univ. Press  (1944)</TD></TR><TR><TD valign="top">[8]</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></table>
+
one has $ \sum _ {n=1} ^ {m} | a _ {n} | ^ {2} \leq \sum _ {n=1} ^ {m} | A _ {n} | ^ {2} $,  
 
+
$ m = 1, 2 \dots $[[#References|[2]]].
 
 
 
 
====Comments====
 
  
 +
The general theory of subordination is useful in considering the set of values taken or produced by an analytic function. The subordination relation can be used in two different ways. First, one can start from a given function  $  F( z) $
 +
and examine the behaviour of all  $  f( z) $
 +
subordinate to  $  F( z) $.
 +
If  $  F( z) $
 +
is completely known, then the region  $  G _ {r} ( F  ) $
 +
in which the values of  $  f( z) $
 +
lie is also known (Theorem 1) as well as an upper bound on the integral means  $  M _  \gamma  ( r, f  ) $(
 +
Theorem 2). If also  $  F( z) $
 +
is regular at  $  z = 0 $,
 +
there are upper bounds for the coefficients of  $  f( z) $(
 +
Theorem 3). Secondly, one can consider a function  $  f( z) $
 +
that is meromorphic in the disc  $  | z | < 1 $
 +
and whose properties imply that it cannot be subordinate to a given function  $  F( z) $
 +
in  $  | z | < 1 $.
 +
If here  $  F( z) $,
 +
for example, is univalent, then  $  f( z) $
 +
necessarily takes values outside  $  G( F  ) $
 +
in  $  | z | < 1 $.
 +
These ideas of using the subordination relation illustrate the subordination principle and can be extended to multiply-connected domains [[#References|[3]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,  "Theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090990/s09099069.png" /> spaces" , Acad. Press  (1970)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.E. Littlewood,  "On inequalities in the theory of functions"  ''Proc. Lond. Math. Soc. (2)'' , '''23'''  (1925)  pp. 481–519</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rogosinski,  "On subordinate functions"  ''Proc. Cambridge Philos. Soc.'' , '''35'''  (1939)  pp. 1–26</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu. Alenitsyn,  "A generalization of the subordination principle to multiply-connected domains"  ''Trudy Mat. Inst. Steklov.'' , '''60'''  (1961)  pp. 5–21  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Rogosinski,  ''Schr. K. Gelehrt. Gesellsch. Naturwiss. Kl.'' , '''8''' :  1  (1931)  pp. 1–31</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Rogosinski,  "On a theorem of Bieberbach–Eilenberg"  ''J. Lond. Math. Soc.'' , '''14''' : 53  (1939)  pp. 4–11</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  W. Rogosinski,  "On the coefficients of subordinate functions"  ''Proc. London Math. Soc.'' , '''48'''  (1943)  pp. 48–82</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.E. Littlewood,  "Lectures on the theory of functions" , Oxford Univ. Press  (1944)</TD></TR><TR><TD valign="top">[8]</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">[a1]</TD> <TD valign="top"> P.L. Duren,  "Theory of $H^p$ spaces" , Acad. Press  (1970)</TD></TR></table>

Latest revision as of 09:10, 6 January 2024


One of the forms of the Lindelöf principle, which employs the concept of subordination of functions. Let $ \omega ( x) $ be any function regular in the disc $ | z | < 1 $ and satisfying the conditions $ \omega ( 0) = 0 $, $ | \omega ( z) | < 1 $ in $ | z | < 1 $; let $ F( z) $ be a meromorphic function in $ | z | < 1 $. If the function $ f( z) $ has the form $ f( z) = F( \omega ( z)) $, then $ f( z) $ is called subordinate to the function $ F( z) $ in the disc $ | z | < 1 $, while $ F( z) $ is called the subordinating function. This subordination relation is denoted by $ f( z) \prec F( z) $. In the simplest case where $ F( z) $ is a univalent function in $ | z | < 1 $, this relation simply means that $ f( 0) = F( 0) $ and that $ f( z) $ does not take any values in the disc $ | z | < 1 $ that are not taken there by $ F( z) $. The following basic theorems apply.

Theorem 1.

Let the function $ w = F( z) $ be meromorphic in the disc $ | z | < 1 $ and map it on the Riemann surface $ G( F ) $. Let $ G _ {r} ( F ) $ be the part of $ G( F ) $ corresponding to $ | z | \leq r $, $ r < 1 $. If $ f( z) \prec F( z) $, then the values of $ f( z) $ in $ | z | \leq r $( under analytic continuation from $ f( 0) = F( 0) $) lie in $ G _ {r} ( F ) $, and the boundary points in $ G _ {r} ( F ) $ are obtained only for $ f( z) = F( \epsilon z) $, $ | \epsilon | = 1 $[2].

Theorem 2.

If $ f( z) \prec F( z) $ and if $ F( z) $ is regular in $ | z | \leq r $, $ r < 1 $, then setting

$$ M _ \lambda ( r, f ) = \ \left \{ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } | f( re ^ {i \theta } ) | ^ \lambda d \phi \right \} ^ {1/ \lambda } ,\ \ \lambda \geq 0 , $$

one has $ M _ \lambda ( r, f ) \leq M _ \lambda ( r, F ) $, $ \lambda \geq 0 $, $ 0 \leq r < 1 $, [1].

Theorem 3.

If $ f( z) \prec F( z) $ and $ F( z) $ is regular at $ z = 0 $, then for the coefficients of the expansions $ f( z) = \sum _ {n=0} ^ \infty a _ {n} z ^ {n} $, $ F( z) = \sum _ {n=0} ^ \infty A _ {n} z ^ {n} $ one has $ \sum _ {n=1} ^ {m} | a _ {n} | ^ {2} \leq \sum _ {n=1} ^ {m} | A _ {n} | ^ {2} $, $ m = 1, 2 \dots $[2].

The general theory of subordination is useful in considering the set of values taken or produced by an analytic function. The subordination relation can be used in two different ways. First, one can start from a given function $ F( z) $ and examine the behaviour of all $ f( z) $ subordinate to $ F( z) $. If $ F( z) $ is completely known, then the region $ G _ {r} ( F ) $ in which the values of $ f( z) $ lie is also known (Theorem 1) as well as an upper bound on the integral means $ M _ \gamma ( r, f ) $( Theorem 2). If also $ F( z) $ is regular at $ z = 0 $, there are upper bounds for the coefficients of $ f( z) $( Theorem 3). Secondly, one can consider a function $ f( z) $ that is meromorphic in the disc $ | z | < 1 $ and whose properties imply that it cannot be subordinate to a given function $ F( z) $ in $ | z | < 1 $. If here $ F( z) $, for example, is univalent, then $ f( z) $ necessarily takes values outside $ G( F ) $ in $ | z | < 1 $. These ideas of using the subordination relation illustrate the subordination principle and can be extended to multiply-connected domains [3].

References

[1] J.E. Littlewood, "On inequalities in the theory of functions" Proc. Lond. Math. Soc. (2) , 23 (1925) pp. 481–519
[2] W. Rogosinski, "On subordinate functions" Proc. Cambridge Philos. Soc. , 35 (1939) pp. 1–26
[3] Yu. Alenitsyn, "A generalization of the subordination principle to multiply-connected domains" Trudy Mat. Inst. Steklov. , 60 (1961) pp. 5–21 (In Russian)
[4] W. Rogosinski, Schr. K. Gelehrt. Gesellsch. Naturwiss. Kl. , 8 : 1 (1931) pp. 1–31
[5] W. Rogosinski, "On a theorem of Bieberbach–Eilenberg" J. Lond. Math. Soc. , 14 : 53 (1939) pp. 4–11
[6] W. Rogosinski, "On the coefficients of subordinate functions" Proc. London Math. Soc. , 48 (1943) pp. 48–82
[7] J.E. Littlewood, "Lectures on the theory of functions" , Oxford Univ. Press (1944)
[8] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[a1] P.L. Duren, "Theory of $H^p$ spaces" , Acad. Press (1970)
How to Cite This Entry:
Subordination principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subordination_principle&oldid=18946
This article was adapted from an original article by Yu.E. Alenitsyn (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article