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Theorems for finding a solution to the [[Coefficient problem|coefficient problem]] for bounded analytic functions. They were obtained by I. Schur [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s0834801.png" /> be the class of regular functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s0834802.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s0834803.png" /> satisfying in it the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s0834804.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s0834805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s0834806.png" />, be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s0834807.png" />-dimensional complex Euclidean space, its points are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s0834808.png" />-tuples of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s0834809.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348010.png" /> be a set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348011.png" /> such that the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348012.png" /> are the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348013.png" /> coefficients of some function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348014.png" />. The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348015.png" /> are closed, bounded and convex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348016.png" />. Then the following theorems hold.
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Schur's first theorem: To the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348017.png" /> on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348018.png" /> there correspond in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348019.png" /> only rational functions of the form
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<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/s083/s083480/s08348020.png" /></td> </tr></table>
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Theorems for finding a solution to the [[Coefficient problem|coefficient problem]] for bounded analytic functions. They were obtained by I. Schur [[#References|[1]]]. Let  $  B $
 +
be the class of regular functions  $  f{ ( z) } = c _ { 0 }  {+c} _ { 1 }  {z+\dots}  $
 +
in  $  | z | < 1 $
 +
satisfying in it the condition  $  | f{ ( z ) } | \leq  1 $.
 +
Let  $  \mathbf C ^ { n } $,
 +
$  n \geq  1 $,
 +
be the  $  n $-dimensional complex Euclidean space, its points are  $  n $-
 +
tuples of complex numbers  $  { ( c _ { 0 },  \dots, c _ {  {n-1}  } ) } $;  
 +
let  $  B ^ { { ( n) } } $
 +
be a set of points  $  { ( c _ { 0 }, \dots, c _ {  {n-1}  } ) } \in \mathbf C ^ { n } $
 +
such that the numbers  $  c _ { 0 }, \dots, c _ {  {n-1}  } $
 +
are the first  $  n $
 +
coefficients of some function from  $  B $.  
 +
The sets  $  B ^ { { ( n) } } $
 +
are closed, bounded and convex in  $  \mathbf C ^ { n } $.  
 +
Then the following theorems hold.
  
Schur's second theorem: A necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348021.png" /> to be an interior point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348022.png" /> is that the following inequalities hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348023.png" />:
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Schur's first theorem: To the points  $  { ( c _ { 0 },  \dots, c _ {  {n-1}  } ) } $
 +
on the boundary of  $  B ^ { { ( n) } } $
 +
there correspond in  $  B $
 +
only rational functions of the form
  
<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/s083/s083480/s08348024.png" /></td> </tr></table>
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$$  \frac{ \overline{\alpha _ {  {n-1}  } } + \overline{\alpha _ {  {n-2}  } }  {z+\dots}  + \overline{\alpha _ { 0 }  } z ^ {  {n-1}  }  }{ \alpha _ { 0 }  {+\alpha} _ { 1 }  {z+\dots} {+\alpha} _ {  {n-1}  } z ^ {  {n-1}  }  }  .  $$
 +
 
 +
Schur's second theorem: A necessary and sufficient condition for  $  { ( c _ { 0 },  \dots, c _ {  {n-1}  } ) } $
 +
to be an interior point of  $  B ^ { { ( n) } } $
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is that the following inequalities hold for  $  k = 1, \dots, n $:
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$$  \left |  \begin{array}{llllllll}  1  & 0  &\cdots  & 0  &c _ { 0 }  &c _ { 1 }  &\cdots  &c _ {  {k-1}  }  \\  0  & 1  &\cdots  & 0  & 0  &c _ { 0 }  &\cdots  &c _ {  {k-2}  }  \\ \vdots  &\vdots  &\ddots  &\vdots  &\vdots  &\vdots  &\ddots  &\vdots  \\  0  & 0  &\cdots  & 1  & 0  & 0  &\cdots  &c _ { 0 }  \\ \overline{c _ { 0 }  }  & 0  &\cdots  & 0  & 1  & 0  &\cdots  & 0  \\ \overline{c _ { 1 }  }  &\overline{c _ { 0 }  }  &\cdots  & 0  & 0  & 1  &\cdots  & 0  \\ \vdots  &\vdots  &\ddots  &\vdots  &\vdots  &\vdots  &\ddots  &\vdots  \\ \overline{c _ {  {k-1}  } }  &\overline{c _ {  {k-2}  } }  &\cdots  &\overline{c _ { 0 }  }  & 0  & 0  &\cdots  & 1  \\ \end{array}  \right |  > 0. $$
  
 
Schur's second theorem provides the complete solution to the coefficient problem for bounded functions in the case of interior points of the coefficients region.
 
Schur's second theorem provides the complete solution to the coefficient problem for bounded functions in the case of interior points of the coefficients region.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Schur,  "Ueber Potentzreihen, die im Innern des Einheitkreises berchränkt sind"  ''J. Reine Angew. Math.'' , '''147'''  (1917)  pp. 205–232</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Bieberbach,  "Lehrbuch der Funktionentheorie" , '''2''' , Teubner  (1931)</TD></TR><TR><TD valign="top">[3]</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>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Schur,  "Ueber Potentzreihen, die im Innern des Einheitkreises berchränkt sind"  ''J. Reine Angew. Math.'' , '''147'''  (1917)  pp. 205–232</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Bieberbach,  "Lehrbuch der Funktionentheorie" , '''2''' , Teubner  (1931)</TD></TR><TR><TD valign="top">[3]</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>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,  "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.B. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)  pp. 40</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,  "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.B. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)  pp. 40</TD></TR></table>

Latest revision as of 00:51, 21 January 2022


Theorems for finding a solution to the coefficient problem for bounded analytic functions. They were obtained by I. Schur [1]. Let $ B $ be the class of regular functions $ f{ ( z) } = c _ { 0 } {+c} _ { 1 } {z+\dots} $ in $ | z | < 1 $ satisfying in it the condition $ | f{ ( z ) } | \leq 1 $. Let $ \mathbf C ^ { n } $, $ n \geq 1 $, be the $ n $-dimensional complex Euclidean space, its points are $ n $- tuples of complex numbers $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } $; let $ B ^ { { ( n) } } $ be a set of points $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } \in \mathbf C ^ { n } $ such that the numbers $ c _ { 0 }, \dots, c _ { {n-1} } $ are the first $ n $ coefficients of some function from $ B $. The sets $ B ^ { { ( n) } } $ are closed, bounded and convex in $ \mathbf C ^ { n } $. Then the following theorems hold.

Schur's first theorem: To the points $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } $ on the boundary of $ B ^ { { ( n) } } $ there correspond in $ B $ only rational functions of the form

$$ \frac{ \overline{\alpha _ { {n-1} } } + \overline{\alpha _ { {n-2} } } {z+\dots} + \overline{\alpha _ { 0 } } z ^ { {n-1} } }{ \alpha _ { 0 } {+\alpha} _ { 1 } {z+\dots} {+\alpha} _ { {n-1} } z ^ { {n-1} } } . $$

Schur's second theorem: A necessary and sufficient condition for $ { ( c _ { 0 }, \dots, c _ { {n-1} } ) } $ to be an interior point of $ B ^ { { ( n) } } $ is that the following inequalities hold for $ k = 1, \dots, n $:

$$ \left | \begin{array}{llllllll} 1 & 0 &\cdots & 0 &c _ { 0 } &c _ { 1 } &\cdots &c _ { {k-1} } \\ 0 & 1 &\cdots & 0 & 0 &c _ { 0 } &\cdots &c _ { {k-2} } \\ \vdots &\vdots &\ddots &\vdots &\vdots &\vdots &\ddots &\vdots \\ 0 & 0 &\cdots & 1 & 0 & 0 &\cdots &c _ { 0 } \\ \overline{c _ { 0 } } & 0 &\cdots & 0 & 1 & 0 &\cdots & 0 \\ \overline{c _ { 1 } } &\overline{c _ { 0 } } &\cdots & 0 & 0 & 1 &\cdots & 0 \\ \vdots &\vdots &\ddots &\vdots &\vdots &\vdots &\ddots &\vdots \\ \overline{c _ { {k-1} } } &\overline{c _ { {k-2} } } &\cdots &\overline{c _ { 0 } } & 0 & 0 &\cdots & 1 \\ \end{array} \right | > 0. $$

Schur's second theorem provides the complete solution to the coefficient problem for bounded functions in the case of interior points of the coefficients region.

References

[1] I. Schur, "Ueber Potentzreihen, die im Innern des Einheitkreises berchränkt sind" J. Reine Angew. Math. , 147 (1917) pp. 205–232
[2] L. Bieberbach, "Lehrbuch der Funktionentheorie" , 2 , Teubner (1931)
[3] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)

Comments

References

[a1] P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
[a2] J.B. Garnett, "Bounded analytic functions" , Acad. Press (1981) pp. 40
How to Cite This Entry:
Schur theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_theorems&oldid=12059
This article was adapted from an original article by Yu.E. Alenitsyn (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article