View source for Rotation theorems

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:

<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>

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.

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

<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>

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):

<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>

<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>

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]]).

====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>



====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]].

====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>