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Various results on conformal and quasi-conformal mappings obtained by H. Grötzsch . He developed the strip method, which is the first general form of the method of conformal moduli (cf. [[Extremal metric, method of the|Extremal metric, method of the]]; [[Strip method (analytic functions)|Strip method (analytic functions)]]), and used it in his systematic study of a large number of extremal problems of conformal mapping of multiply-connected (including infinitely-connected) domains, including the problems of the existence, uniqueness and geometric properties of extremal mappings. A few of the simpler Grötzsch theorems are presented below.
 
Various results on conformal and quasi-conformal mappings obtained by H. Grötzsch . He developed the strip method, which is the first general form of the method of conformal moduli (cf. [[Extremal metric, method of the|Extremal metric, method of the]]; [[Strip method (analytic functions)|Strip method (analytic functions)]]), and used it in his systematic study of a large number of extremal problems of conformal mapping of multiply-connected (including infinitely-connected) domains, including the problems of the existence, uniqueness and geometric properties of extremal mappings. A few of the simpler Grötzsch theorems are presented below.
  
Of all univalent conformal mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452001.png" /> of a given annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452002.png" /> under which the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452003.png" /> is mapped onto itself, the maximum diameter of the image of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452004.png" /> is attained if and only if the boundary component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452005.png" /> is a rectilinear segment with its centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452006.png" />. A similar result is valid for multiply-connected domains.
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Of all univalent conformal mappings $  w = f ( z) $
 +
of a given annulus $  K _ {R} = \{ {z } : {R < | z | < 1 } \} $
 +
under which the unit circle $  \Gamma = \{ {z } : {| z | = 1 } \} $
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is mapped onto itself, the maximum diameter of the image of the circle $  \Gamma _ {R} = \{ {z } : {| z | = R } \} $
 +
is attained if and only if the boundary component $  f ( \Gamma _ {R} ) $
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is a rectilinear segment with its centre at the point $  w = 0 $.  
 +
A similar result is valid for multiply-connected domains.
  
Out of all univalent conformal mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452007.png" /> of a given multiply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452008.png" /> with expansion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g0452009.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520010.png" /> at infinity and normalization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520011.png" /> at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520012.png" />, the maximum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520013.png" />, and the maximum (minimum) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520014.png" /> at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520016.png" />, are attained only on mappings that map each boundary component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520017.png" />, respectively, to an arc of a circle with centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520018.png" />, or to an arc of an ellipse (hyperbola) with foci at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520020.png" />. In each one of these problems the extremal mapping exists and is unique. In this class of mappings, for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520021.png" />, the disc
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Out of all univalent conformal mappings $  w = f ( z) $
 +
of a given multiply-connected domain $  B \ni \infty $
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with expansion $  f ( z) = z + O ( 1) $
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$  ( z \rightarrow \infty ) $
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at infinity and normalization $  f ( z _ {0} ) = 0 $
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at a given point $  z _ {0} \in B $,  
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the maximum of $  | f ^ { \prime } ( z _ {0} ) | $,  
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and the maximum (minimum) of $  | f ( z _ {1} ) | $
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at a given point $  z _ {1} \in B $,  
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$  z _ {1} \neq z _ {0} $,  
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are attained only on mappings that map each boundary component of $  B $,  
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respectively, to an arc of a circle with centre at the point $  w = 0 $,  
 +
or to an arc of an ellipse (hyperbola) with foci at the points $  w = 0 $
 +
and $  w = w  ^  \prime  = f ( z _ {1} ) $.  
 +
In each one of these problems the extremal mapping exists and is unique. In this class of mappings, for a given $  z _ {1} \in B $,  
 +
the disc
  
<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/g/g045/g045200/g04520022.png" /></td> </tr></table>
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$$
 +
\left \{ {w } : {
 +
\left |
 +
w - {
 +
\frac{1}{2}
 +
} ( w  ^  \prime  + w  ^ {\prime\prime} ) \
 +
\right |  \leq  \
 +
{
 +
\frac{1}{2}
 +
} | w  ^  \prime  - w  ^ {\prime\prime} | \
 +
} \right \}
 +
$$
  
is the range of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520023.png" />. Each boundary point of this disc is a value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045200/g04520024.png" /> on a unique mapping in the class under study with specific geometric properties.
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is the range of the function $  \Phi ( f  ) = \mathop{\rm ln} ( f ( z _ {1} )/z _ {1} ) $.  
 +
Each boundary point of this disc is a value of $  \Phi $
 +
on a unique mapping in the class under study with specific geometric properties.
  
 
Grötzsch was the first to propose a form of representation of a [[Quasi-conformal mapping|quasi-conformal mapping]], and to apply to such a mappings many extremal results which he had formerly obtained for conformal mappings.
 
Grötzsch was the first to propose a form of representation of a [[Quasi-conformal mapping|quasi-conformal mapping]], and to apply to such a mappings many extremal results which he had formerly obtained for conformal mappings.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  H. Grötzsch,  "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche I"  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl.'' , '''81'''  (1929)  pp. 38–47</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  H. Grötzsch,  "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche II"  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl.'' , '''81'''  (1929)  pp. 217–221</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  H. Grötzsch,  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl.'' , '''82'''  (1930)  pp. 69–80</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  H. Grötzsch,  "Ueber die Verschiebung bei schlichter konformer Abbildung schlichter Bereiche II"  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl.'' , '''84'''  (1932)  pp. 269–278</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  H. Grötzsch,  "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche I"  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl.'' , '''81'''  (1929)  pp. 38–47</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  H. Grötzsch,  "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche II"  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl.'' , '''81'''  (1929)  pp. 217–221</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  H. Grötzsch,  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl.'' , '''82'''  (1930)  pp. 69–80</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top">  H. Grötzsch,  "Ueber die Verschiebung bei schlichter konformer Abbildung schlichter Bereiche II"  ''Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl.'' , '''84'''  (1932)  pp. 269–278</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:42, 5 June 2020


Various results on conformal and quasi-conformal mappings obtained by H. Grötzsch . He developed the strip method, which is the first general form of the method of conformal moduli (cf. Extremal metric, method of the; Strip method (analytic functions)), and used it in his systematic study of a large number of extremal problems of conformal mapping of multiply-connected (including infinitely-connected) domains, including the problems of the existence, uniqueness and geometric properties of extremal mappings. A few of the simpler Grötzsch theorems are presented below.

Of all univalent conformal mappings $ w = f ( z) $ of a given annulus $ K _ {R} = \{ {z } : {R < | z | < 1 } \} $ under which the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $ is mapped onto itself, the maximum diameter of the image of the circle $ \Gamma _ {R} = \{ {z } : {| z | = R } \} $ is attained if and only if the boundary component $ f ( \Gamma _ {R} ) $ is a rectilinear segment with its centre at the point $ w = 0 $. A similar result is valid for multiply-connected domains.

Out of all univalent conformal mappings $ w = f ( z) $ of a given multiply-connected domain $ B \ni \infty $ with expansion $ f ( z) = z + O ( 1) $ $ ( z \rightarrow \infty ) $ at infinity and normalization $ f ( z _ {0} ) = 0 $ at a given point $ z _ {0} \in B $, the maximum of $ | f ^ { \prime } ( z _ {0} ) | $, and the maximum (minimum) of $ | f ( z _ {1} ) | $ at a given point $ z _ {1} \in B $, $ z _ {1} \neq z _ {0} $, are attained only on mappings that map each boundary component of $ B $, respectively, to an arc of a circle with centre at the point $ w = 0 $, or to an arc of an ellipse (hyperbola) with foci at the points $ w = 0 $ and $ w = w ^ \prime = f ( z _ {1} ) $. In each one of these problems the extremal mapping exists and is unique. In this class of mappings, for a given $ z _ {1} \in B $, the disc

$$ \left \{ {w } : { \left | w - { \frac{1}{2} } ( w ^ \prime + w ^ {\prime\prime} ) \ \right | \leq \ { \frac{1}{2} } | w ^ \prime - w ^ {\prime\prime} | \ } \right \} $$

is the range of the function $ \Phi ( f ) = \mathop{\rm ln} ( f ( z _ {1} )/z _ {1} ) $. Each boundary point of this disc is a value of $ \Phi $ on a unique mapping in the class under study with specific geometric properties.

Grötzsch was the first to propose a form of representation of a quasi-conformal mapping, and to apply to such a mappings many extremal results which he had formerly obtained for conformal mappings.

References

[1a] H. Grötzsch, "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche I" Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 81 (1929) pp. 38–47
[1b] H. Grötzsch, "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche II" Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 81 (1929) pp. 217–221
[1c] H. Grötzsch, Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 82 (1930) pp. 69–80
[1d] H. Grötzsch, "Ueber die Verschiebung bei schlichter konformer Abbildung schlichter Bereiche II" Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 84 (1932) pp. 269–278
[2] J.A. Jenkins, "Univalent functions and conformal mappings" , Springer (1958)

Comments

Grötzsch' theorems are distortion theorems.

Cf. also Grötzsch principle.

How to Cite This Entry:
Grötzsch theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gr%C3%B6tzsch_theorems&oldid=17095
This article was adapted from an original article by P.M. Tamrazov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article