Rotation theorems

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:

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