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Theorems which characterize the change in the argument under a [[Conformal mapping|conformal mapping]]. Rotation theorems in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r0826801.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r0826802.png" /> which are regular and univalent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r0826803.png" /> give accurate estimates of the argument of the derivative for functions of this class:
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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/r/r082/r082680/r0826804.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
Here one considers the branch of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r0826805.png" /> that vanishes when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r0826806.png" />. The upper and the lower bounds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r0826807.png" /> given by the inequalities (*) are sharp for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r0826808.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r0826809.png" />. This rotation theorem was obtained by G.M. Goluzin [[#References|[1]]], [[#References|[5]]]; I.E. Bazilevich [[#References|[2]]] was the first to show that the inequalities (*) are sharp for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r08268010.png" />; J.A. Jenkins [[#References|[3]]] gave a complete analysis of the cases of equality in these estimates.
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Theorems which characterize the change in the argument under a [[Conformal mapping|conformal mapping]]. Rotation theorems in the class  $  S $
 +
of functions  $  f( z) = z + c _ {2} z  ^ {2} + \dots $
 +
which are regular and univalent in the disc  $  | z | < 1 $
 +
give accurate estimates of the argument of the derivative for functions of this class:
  
Rotation theorems in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r08268012.png" /> is also the name given to estimates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r08268013.png" /> and to estimates of expressions of the type
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$$ \tag{* }
 +
|  \mathop{\rm arg}  f ^ { \prime } ( z) |  \leq  \left \{
  
<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/r/r082/r082680/r08268014.png" /></td> </tr></table>
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\begin{array}{ll}
 +
4  \mathop{\rm arc}  \sin  | z |  &\textrm{ if }  | z | \leq  2 ^ {- 1/2 } ,  \\
 +
\pi +  \mathop{\rm ln} \
  
The simplest estimates of this type in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r08268015.png" /> are the sharp inequalities (the appropriate branches of the arguments are considered):
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\frac{| z |  ^ {2} }{1 - | z |  ^ {2} }
 +
  &\textrm{ if }  2 ^ {- 1/2 } \leq  | z | < 1. \\
 +
\end{array}
  
<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/r/r082/r082680/r08268016.png" /></td> </tr></table>
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\right .$$
  
<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/r/r082/r082680/r08268017.png" /></td> </tr></table>
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Here one considers the branch of  $  \mathop{\rm arg}  f ^ { \prime } ( z) $
 +
that vanishes when  $  z = 0 $.
 +
The upper and the lower bounds for  $  \mathop{\rm arg}  f ^ { \prime } ( z) $
 +
given by the inequalities (*) are sharp for any  $  z $
 +
in the disc  $  | z | < 1 $.
 +
This rotation theorem was obtained by G.M. Goluzin [[#References|[1]]], [[#References|[5]]]; I.E. Bazilevich [[#References|[2]]] was the first to show that the inequalities (*) are sharp for  $  2 ^ {- 1/2 } < | z | < 1 $;  
 +
J.A. Jenkins [[#References|[3]]] gave a complete analysis of the cases of equality in these estimates.
  
There are also rotation theorems in other classes of functions which realize a univalent conformal mapping of the disc or its exterior, and in classes of functions which are univalent in a multiply-connected domain (cf. [[#References|[5]]], [[#References|[3]]], [[Distortion theorems|Distortion theorems]]; [[Univalent function|Univalent function]]). Rotation theorems have also been extended to include the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r08268018.png" />-valued functions (cf. addenda to [[#References|[5]]], and also [[Multivalent function|Multivalent function]]).
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Rotation theorems in the class  $  S $
 +
is also the name given to estimates of  $  \mathop{\rm arg} ( f( z)/z) $
 +
and to estimates of expressions of the type
 +
 
 +
$$
 +
\lambda  \mathop{\rm arg}  f ^ { \prime } ( z) - ( 1 - \lambda )  \mathop{\rm arg} \
 +
 
 +
\frac{f ( z) }{z }
 +
,\ \
 +
0 < \lambda < 1.
 +
$$
 +
 
 +
The simplest estimates of this type in the class  $  S $
 +
are the sharp inequalities (the appropriate branches of the arguments are considered):
 +
 
 +
$$
 +
\left |  \mathop{\rm arg} 
 +
\frac{f( z) }{z }
 +
\right |  \leq    \mathop{\rm ln} \
 +
 
 +
\frac{1 + | z | }{1 - | z | }
 +
,\  | z | < 1;
 +
$$
 +
 
 +
$$
 +
\left |  \mathop{\rm arg} 
 +
\frac{zf ^ { \prime } ( z) }{f( z) }
 +
\right |
 +
\leq    \mathop{\rm ln} 
 +
\frac{1 + | z | }{1 - | z | }
 +
,\ \
 +
| z | < 1.
 +
$$
 +
 
 +
There are also rotation theorems in other classes of functions which realize a univalent conformal mapping of the disc or its exterior, and in classes of functions which are univalent in a multiply-connected domain (cf. [[#References|[5]]], [[#References|[3]]], [[Distortion theorems|Distortion theorems]]; [[Univalent function|Univalent function]]). Rotation theorems have also been extended to include the case of $  p $-
 +
valued functions (cf. addenda to [[#References|[5]]], and also [[Multivalent function|Multivalent function]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,  "On distortion theorems in the theory of conformal mappings"  ''Mat. Sb.'' , '''1 (43)''' :  1  (1936)  pp. 127–135  (In Russian)  (German abstract)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.E. Bazilevich,  "Sur les théorèmes de Koebe–Bieberbach"  ''Mat. Sb.'' , '''1 (43)''' :  3  (1936)  pp. 283–292</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Grunsky,  "Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche"  ''Schriftenreihe Math. Sem. Inst. Angew. Math. Univ. Berlin'' , '''1'''  (1932)  pp. 95–140</TD></TR><TR><TD valign="top">[5]</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">  G.M. Goluzin,  "On distortion theorems in the theory of conformal mappings"  ''Mat. Sb.'' , '''1 (43)''' :  1  (1936)  pp. 127–135  (In Russian)  (German abstract)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.E. Bazilevich,  "Sur les théorèmes de Koebe–Bieberbach"  ''Mat. Sb.'' , '''1 (43)''' :  3  (1936)  pp. 283–292</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Jenkins,  "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Grunsky,  "Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche"  ''Schriftenreihe Math. Sem. Inst. Angew. Math. Univ. Berlin'' , '''1'''  (1932)  pp. 95–140</TD></TR><TR><TD valign="top">[5]</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====
For the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082680/r08268019.png" /> see also [[Bieberbach conjecture|Bieberbach conjecture]].
+
For the class $  S $
 +
see also [[Bieberbach conjecture|Bieberbach conjecture]].
  
 
====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></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,  "Univalent functions" , Springer  (1983)  pp. Sect. 10.11</TD></TR></table>

Revision as of 14:55, 7 June 2020


Theorems which characterize the change in the argument under a conformal mapping. Rotation theorems in the class $ S $ of functions $ f( z) = z + c _ {2} z ^ {2} + \dots $ which are regular and univalent in the disc $ | z | < 1 $ give accurate estimates of the argument of the derivative for functions of this class:

$$ \tag{* } | \mathop{\rm arg} f ^ { \prime } ( z) | \leq \left \{ \begin{array}{ll} 4 \mathop{\rm arc} \sin | z | &\textrm{ if } | z | \leq 2 ^ {- 1/2 } , \\ \pi + \mathop{\rm ln} \ \frac{| z | ^ {2} }{1 - | z | ^ {2} } &\textrm{ if } 2 ^ {- 1/2 } \leq | z | < 1. \\ \end{array} \right .$$

Here one considers the branch of $ \mathop{\rm arg} f ^ { \prime } ( z) $ that vanishes when $ z = 0 $. The upper and the lower bounds for $ \mathop{\rm arg} f ^ { \prime } ( z) $ given by the inequalities (*) are sharp for any $ z $ in the disc $ | z | < 1 $. This rotation theorem was obtained by G.M. Goluzin [1], [5]; I.E. Bazilevich [2] was the first to show that the inequalities (*) are sharp for $ 2 ^ {- 1/2 } < | z | < 1 $; J.A. Jenkins [3] gave a complete analysis of the cases of equality in these estimates.

Rotation theorems in the class $ S $ is also the name given to estimates of $ \mathop{\rm arg} ( f( z)/z) $ and to estimates of expressions of the type

$$ \lambda \mathop{\rm arg} f ^ { \prime } ( z) - ( 1 - \lambda ) \mathop{\rm arg} \ \frac{f ( z) }{z } ,\ \ 0 < \lambda < 1. $$

The simplest estimates of this type in the class $ S $ are the sharp inequalities (the appropriate branches of the arguments are considered):

$$ \left | \mathop{\rm arg} \frac{f( z) }{z } \right | \leq \mathop{\rm ln} \ \frac{1 + | z | }{1 - | z | } ,\ | z | < 1; $$

$$ \left | \mathop{\rm arg} \frac{zf ^ { \prime } ( z) }{f( z) } \right | \leq \mathop{\rm ln} \frac{1 + | z | }{1 - | z | } ,\ \ | z | < 1. $$

There are also rotation theorems in other classes of functions which realize a univalent conformal mapping of the disc or its exterior, and in classes of functions which are univalent in a multiply-connected domain (cf. [5], [3], Distortion theorems; Univalent function). Rotation theorems have also been extended to include the case of $ p $- valued functions (cf. addenda to [5], and also Multivalent function).

References

[1] G.M. Goluzin, "On distortion theorems in the theory of conformal mappings" Mat. Sb. , 1 (43) : 1 (1936) pp. 127–135 (In Russian) (German abstract)
[2] I.E. Bazilevich, "Sur les théorèmes de Koebe–Bieberbach" Mat. Sb. , 1 (43) : 3 (1936) pp. 283–292
[3] J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)
[4] H. Grunsky, "Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche" Schriftenreihe Math. Sem. Inst. Angew. Math. Univ. Berlin , 1 (1932) pp. 95–140
[5] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)

Comments

For the class $ S $ see also Bieberbach conjecture.

References

[a1] P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
How to Cite This Entry:
Rotation theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rotation_theorems&oldid=49412
This article was adapted from an original article by G.V. Kuz'mina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article