# 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).

#### 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)

For the class $S$ see also Bieberbach conjecture.