Namespaces
Variants
Actions

Difference between revisions of "Schur theorems"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (using vdots and ddots)
 
(3 intermediate revisions by one other user not shown)
Line 12: Line 12:
  
 
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 $
 
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 $
+
be the class of regular functions  $  f{ ( z) } = c _ { 0 } {+c} _ { 1 }   {z+\dots$
 
in  $  | z | < 1 $
 
in  $  | z | < 1 $
satisfying in it the condition  $  | f( z ) | \leq  1 $.  
+
satisfying in it the condition  $  | f{ ( z ) } | \leq  1 $.  
Let  $  \mathbf C ^ {n} $,  
+
Let  $  \mathbf C ^ { n } $,  
 
$  n \geq  1 $,  
 
$  n \geq  1 $,  
be the  $  n $-
+
be the  $  n $-dimensional complex Euclidean space, its points are  $  n $-
dimensional complex Euclidean space, its points are  $  n $-
+
tuples of complex numbers  $  { ( c _ { 0 }\dots, c _ {n-1}  } ) } $;  
tuples of complex numbers  $  ( c _ {0} \dots c _ {n-} 1 ) $;  
+
let  $  B ^ { { ( n) } } $
let  $  B ^ {(} n) $
+
be a set of points  $  { ( c _ { 0 }, \dots, c _ {n-1}  } ) } \in \mathbf C ^ { 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}  } $
such that the numbers  $  c _ {0} \dots c _ {n-} 1 $
 
 
are the first  $  n $
 
are the first  $  n $
 
coefficients of some function from  $  B $.  
 
coefficients of some function from  $  B $.  
The sets  $  B ^ {(} n) $
+
The sets  $  B ^ { { ( n) } } $
are closed, bounded and convex in  $  \mathbf C ^ {n} $.  
+
are closed, bounded and convex in  $  \mathbf C ^ { n } $.  
 
Then the following theorems hold.
 
Then the following theorems hold.
  
Schur's first theorem: To the points  $  ( c _ {0} \dots c _ {n-} 1 ) $
+
Schur's first theorem: To the points  $  { ( c _ { 0 }\dots, c _ {n-1}  } ) } $
on the boundary of  $  B ^ {(} n) $
+
on the boundary of  $  B ^ { { ( n) } } $
 
there correspond in  $  B $
 
there correspond in  $  B $
 
only rational functions of the form
 
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}  }  }  .  $$
  
\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 $:
  
Schur's second theorem: A necessary and sufficient condition for $ ( c _ {0} \dots c _ {n-} 1 ) $
+
$$  \left |  \begin{array}{llllllll}  1  & 0  &\cdots  & 0  &c _ { 0 }  &c _ { 1 }  &\cdots  &c _ {  {k-1}  }  \\  0  & 1  &\cdots & 0  & 0 &c _ { 0 }   &\cdots  &c _ { {k-2}  }  \\ \vdots  &\vdots  &\ddots  &\vdots  &\vdots  &\vdots  &\ddots  &\vdots  \\  0  & 0  &\cdots  & 1  & 0  & 0  &\cdots  &c _ { 0 }  \\ \overline{c _ { 0 }  }  & 0  &\cdots  & 0  & 1  & 0  &\cdots  & 0  \\ \overline{c _ { 1 }  }  &\overline{c _ { 0 }  } &\cdots  & 0  & 0  & &\cdots  & 0  \\ \vdots  &\vdots  &\ddots  &\vdots  &\vdots  &\vdots  &\ddots &\vdots  \\ \overline{c _ { {k-1}  } &\overline{c _ { {k-2}  } }  &\cdots  &\overline{c _ { 0 }  }  & 0  & 0  &\cdots  & 1 \\ \end{array}  \right |  >  0.  $$
to be an interior point of $ B ^ {(} n) $
 
is that the following inequalities hold for $ k = 1 \dots n $:
 
 
 
$$
 
\left |
 
  
 
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.

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=48627
This article was adapted from an original article by Yu.E. Alenitsyn (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article