Difference between revisions of "Schur theorems"

Revision as of 14:53, 7 June 2020

Theorems for finding a solution to the coefficient problem for bounded analytic functions. They were obtained by I. Schur [1]. Let be the class of regular functions in satisfying in it the condition . Let , , be the -dimensional complex Euclidean space, its points are -tuples of complex numbers ; let be a set of points such that the numbers are the first coefficients of some function from . The sets are closed, bounded and convex in . Then the following theorems hold.

Schur's first theorem: To the points on the boundary of there correspond in only rational functions of the form

Schur's second theorem: A necessary and sufficient condition for to be an interior point of is that the following inequalities hold for :

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

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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.
-s0834801.png
-$#A+1 = 24 n = 0
-$#C+1 = 24 : ~/encyclopedia/old_files/data/S083/S.0803480 Schur theorems
-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}}
+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
-{{TEX|done}}
-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 $
+<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>
-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 ) $
+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" />:
-on the boundary of  $  B  ^ {(} n) $
-there correspond in  $  B $
-only rational functions of the form
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083480/s08348024.png" /></td> </tr></table>
-\frac{\overline{ {\alpha _ {n-} 1 }}\; + \overline{ {\alpha _ {n-} 2 }}\; z + \dots + \overline{ {\alpha _ {0} }}\; z  ^ {n-} 1 }{\alpha _ {0} + \alpha _ {1} z + \dots + \alpha _ {n-} 1 z  ^ {n-} 1 }
- .
-$$
-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 |
 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.
@@ Line 51: / Line 13: @@
 ====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>
 ====Comments====
 ====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>

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Difference between revisions of "Schur theorems"

Revision as of 14:53, 7 June 2020

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