# Grötzsch theorems

Various results on conformal and quasi-conformal mappings obtained by H. Grötzsch . He developed the strip method, which is the first general form of the method of conformal moduli (cf. Extremal metric, method of the; Strip method (analytic functions)), and used it in his systematic study of a large number of extremal problems of conformal mapping of multiply-connected (including infinitely-connected) domains, including the problems of the existence, uniqueness and geometric properties of extremal mappings. A few of the simpler Grötzsch theorems are presented below.

Of all univalent conformal mappings $w = f ( z)$ of a given annulus $K _ {R} = \{ {z } : {R < | z | < 1 } \}$ under which the unit circle $\Gamma = \{ {z } : {| z | = 1 } \}$ is mapped onto itself, the maximum diameter of the image of the circle $\Gamma _ {R} = \{ {z } : {| z | = R } \}$ is attained if and only if the boundary component $f ( \Gamma _ {R} )$ is a rectilinear segment with its centre at the point $w = 0$. A similar result is valid for multiply-connected domains.

Out of all univalent conformal mappings $w = f ( z)$ of a given multiply-connected domain $B \ni \infty$ with expansion $f ( z) = z + O ( 1)$ $( z \rightarrow \infty )$ at infinity and normalization $f ( z _ {0} ) = 0$ at a given point $z _ {0} \in B$, the maximum of $| f ^ { \prime } ( z _ {0} ) |$, and the maximum (minimum) of $| f ( z _ {1} ) |$ at a given point $z _ {1} \in B$, $z _ {1} \neq z _ {0}$, are attained only on mappings that map each boundary component of $B$, respectively, to an arc of a circle with centre at the point $w = 0$, or to an arc of an ellipse (hyperbola) with foci at the points $w = 0$ and $w = w ^ \prime = f ( z _ {1} )$. In each one of these problems the extremal mapping exists and is unique. In this class of mappings, for a given $z _ {1} \in B$, the disc

$$\left \{ {w } : { \left | w - { \frac{1}{2} } ( w ^ \prime + w ^ {\prime\prime} ) \ \right | \leq \ { \frac{1}{2} } | w ^ \prime - w ^ {\prime\prime} | \ } \right \}$$

is the range of the function $\Phi ( f ) = \mathop{\rm ln} ( f ( z _ {1} )/z _ {1} )$. Each boundary point of this disc is a value of $\Phi$ on a unique mapping in the class under study with specific geometric properties.

Grötzsch was the first to propose a form of representation of a quasi-conformal mapping, and to apply to such a mappings many extremal results which he had formerly obtained for conformal mappings.

#### References

 [1a] H. Grötzsch, "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche I" Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 81 (1929) pp. 38–47 [1b] H. Grötzsch, "Ueber die Verzerrung bei schlichter konformer Abbildung mehrfach zusammenhängender Bereiche II" Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 81 (1929) pp. 217–221 [1c] H. Grötzsch, Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 82 (1930) pp. 69–80 [1d] H. Grötzsch, "Ueber die Verschiebung bei schlichter konformer Abbildung schlichter Bereiche II" Ber. Verh. Sächsisch. Akad. Wiss. Leipzig. Math.-Naturwiss. Kl. , 84 (1932) pp. 269–278 [2] J.A. Jenkins, "Univalent functions and conformal mappings" , Springer (1958)