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:

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

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).

Comments

For the class $S$, see also Bieberbach conjecture.

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)
[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11