# Distortion theorems

under conformal mapping of planar domains

Theorems characterizing the distortion of line elements at a given point of a domain, as well as the distortion of the domain and its subsets, and the distortion of the boundary of the domain under a conformal mapping. Estimates of the modulus of the derivatives of an analytic function at a point of a domain belong first of all to distortion theorems. The statement, for functions in the class $\Sigma$ of functions

$$F ( \zeta ) = \zeta + \alpha _ {0} + \frac{\alpha _ {1} } \zeta + \dots ,$$

meromorphic and univalent in $| \zeta | > 1$, that for all $\zeta _ {0}$, $1 < | \zeta _ {0} | < \infty$, the inequality

$$\tag{1 } 1 - \frac{1}{| \zeta _ {0} | ^ {2} } \leq | F ^ { \prime } ( \zeta _ {0} ) | \leq \ \frac{| \zeta _ {0} | ^ {2} }{| \zeta _ {0} | ^ {2} - 1 }$$

holds, is a distortion theorem.

Equality at the left-hand side of (1) holds only for the functions

$$F _ {1} ( \zeta ) = \zeta + \alpha _ {0} + \zeta _ {0} ( \overline \zeta_ {0} \zeta ) ^ {- 1} ,$$

while at the right-hand side equality holds only for the functions

$$F _ {2} ( \zeta ) = \ \frac{\zeta - \zeta _ {0} }{1 - ( \overline \zeta_ {0} \zeta ) ^ {- 1} } + \beta _ {0} .$$

Here $\alpha _ {0}$ and $\beta _ {0}$ are two arbitrary fixed numbers. The functions $w = F _ {1} ( \zeta )$ map the domain $| \zeta | > 1$ onto the $w$-plane with slit along the interval connecting the points $\alpha _ {0} - 2 \zeta _ {0} / | \zeta _ {0} |$ and $\alpha _ {0} + 2 \zeta _ {0} / | \zeta _ {0} |$. The functions $w = F _ {2} ( \zeta )$ map the domain $| \zeta | > 1$ onto the $w$-plane with slit along an arc of the circle $| w- \beta _ {0} | = | \zeta _ {0} |$ with mid-point $\beta _ {0} - \zeta _ {0}$. Inequality (1) is easily obtained from the Grunsky inequality

$$| \ln F ^ { \prime } ( \zeta _ {0} ) | \leq \ - \ln \left ( 1 - \frac{1}{| \zeta _ {0} | ^ {2} } \right ) ,$$

which determines the range of values of the functional $\ln F ^ { \prime } ( \zeta _ {0} )$ on the class $\Sigma$. On the other hand, inequality (1) is a direct consequence of Goluzin's theorem: If $F ( \zeta ) \in \Sigma$, then for any two points $\zeta _ {1} , \zeta _ {2}$ with $| \zeta _ {1} | = | \zeta _ {2} | = \rho$, $1 < \rho < \infty$, the sharp inequality

$$\tag{2 } \left | \ln \frac{F ( \zeta _ {1} ) - F ( \zeta _ {0} ) }{\zeta _ {1} - \zeta _ {2} } \right | \leq - \ln \left ( 1 - \frac{1}{\rho ^ {2} } \right )$$

holds, where, moreover, the equality sign is attained for the functions $F ( \zeta ) = \zeta + e ^ {i \alpha } / \zeta$, where $\alpha$ is a real constant. Inequality (2) also implies the chord-distortion theorem (cf. [1]). If $F ( \zeta ) \in \Sigma$, then for any two points $\zeta _ {1} , \zeta _ {2}$ on the circle $| \zeta | = \rho > 1$ the sharp inequality

$$\left | \frac{F ( \zeta _ {1} ) - F ( \zeta _ {2} ) }{\zeta _ {1} - \zeta _ {2} } \ \right | \geq \ 1 - \frac{1}{\rho ^ {2} }$$

holds. Equality in this case is only attained for the functions

$$F ( \zeta ) = \zeta + C + \frac{e ^ {2 i \phi } } \zeta ,$$

where $C$ is a constant and $\phi = ( \mathop{\rm arg} \zeta _ {1} + \mathop{\rm arg} \zeta _ {2} ) / 2$. Various generalizations of (2) are known. These give the ranges of values of corresponding functionals and are sharpened versions of distortion theorems for $\Sigma$ or its subclasses (cf., e.g., [1]).

In the class $S$ of functions

$$f ( z) = z + c _ {2} z ^ {2} + \dots$$

that are regular and univalent in the disc $| z | < 1$, the following sharp inequalities are valid for $0 < | z _ {0} | < 1$:

$$\tag{3 } \frac{1 - | z _ {0} | }{( 1 + | z _ {0} | ) ^ {3} } \leq | f ^ { \prime } ( z _ {0} ) | \leq \ \frac{1 + | z _ {0} | }{( 1 - | z _ {0} | ) ^ {3} } ,$$

$$\tag{4 } \frac{| z _ {0} | }{( 1 + | z _ {0} | ) ^ {2} } \leq | f ( z _ {0} ) | \leq \frac{| z _ {0} | }{( 1 - | z _ {0} | ) ^ {2} } ,$$

$$\tag{5 } \frac{1 - | z _ {0} | }{1 + | z _ {0} | } \leq \left | \frac{z _ {0} f ^ { \prime } ( z _ {0} ) }{f ( z _ {0} ) } \right | \leq \frac{1 + | z _ {0} | }{1 - | z _ {0} | } .$$

The estimates (4) and (5) follow from (3). The inequalities (3)–(5) are called the distortion theorems for $S$. The lower bounds are realized only by the functions

$$f _ \alpha ( z) = \ \frac{z}{( 1 + e ^ {- i \alpha } z ) ^ {2} } ,$$

while the upper bounds are realized only by the functions

$$f _ {\pi + \alpha } ( z) = \ \frac{z}{( 1 - e ^ {- i \alpha } z ) ^ {2} } ,$$

where $\alpha = \mathop{\rm arg} z _ {0}$. The functions $w = f _ \alpha ( z)$, $0 \leq \alpha < 2 \pi$, known as the Koebe functions, map the disc $| z | < 1$ onto the $w$-plane with slit along the ray $\mathop{\rm arg} w = \alpha$, $| w | \geq 1 / 4$. They are extremal in a number of problems in the theory of univalent functions. Koebe's $1 / 4$-theorem holds: The domain that is the image of the disc $| z | < 1$ under a mapping $w = f ( z)$, $f \in S$, always contains the disc $| w | < 1 / 4$, and the point $w = e ^ {i \alpha } / 4$ lies on the boundary of this domain only for $f ( z) = f _ \alpha ( z )$.

The estimates (3)–(5) are simple consequences of results on the ranges of the functionals

$$\ln f ^ { \prime } ( z _ {0} ) ,\ \ \ln \frac{f ( z _ {0} ) }{z _ {0} } ,\ \ \ln \frac{z f ^ { \prime } ( z _ {0} ) }{f ( z _ {0} ) }$$

on $S$( cf. [2]).

Let $\Sigma _ {0}$ be the class of functions $F ( \zeta ) \in \Sigma$ with $F ( \zeta ) \neq 0$ for $1 < | \zeta | < \infty$. Between functions in $S$ and $\Sigma _ {0}$ there is the following relation: If $f ( z) \in S$, then $F ( \zeta ) = 1 / f ( 1 / \zeta ) \in \Sigma _ {0}$, and, conversely, if $F ( \zeta ) \in \Sigma _ {0}$, then $f ( z) = 1 / F ( 1 / z ) \in S$. Hence, the range of some functional (or system of functionals) on $S$ is determined by the range of the corresponding functional (system of functionals) on $\Sigma _ {0}$, vice versa. E.g., the range of $\ln f ( z _ {0} ) / z _ {0}$, $0 < | z _ {0} | < 1$, on $S$ is easily obtained from that of $\ln F ( \zeta _ {0} ) / \zeta _ {0}$, $1 < | \zeta _ {0} | < \infty$, on $\Sigma _ {0}$.

For functions that are regular and bounded in a disc, the Schwarz lemma (cf. [1]) and its generalizations, as well as the following boundary-distortion theorem of Löwner are examples of distortion theorems. Löwner's theorem: For a function $\phi ( z)$ that is regular in $| z | < 1$ with $\phi ( 0) = 0$, $| \phi ( z) | < 1$ in $| z | < 1$ and $| \phi ( z) | = 1$ on an arc $A$ of $| z | = 1$, the length of the image of $A$ is not smaller than the length of $A$ itself, and equality only holds for the functions $\phi ( z) = e ^ {i \alpha } z$, with $\alpha$ a real number.

In the class of functions that are univalent in a given multiply-connected domain, the minimum (respectively, maximum) modulus of the derivative at a given point of the domain is attained only for mappings of this domain onto a domain with radial (resp. circular concentric) slits. For unbounded mappings the following theorem holds: Let $D$ be a finitely-connected domain in the $\zeta$-plane containing the point at infinity, let $\Sigma ( D)$ be the class of univalent functions $F ( \zeta )$ in $D$ that have in a neighbourhood of $\zeta = \infty$ the expansion

$$F ( \zeta ) = \zeta + \alpha _ {0} + \frac{\alpha _ {1} } \zeta + \dots ,$$

and let $\zeta _ {0} \neq \infty$ be a point in $D$. Let $F _ \theta ( \zeta )$, $F _ \theta ( \zeta _ {0} ) = 0$, be a function in $\Sigma ( D)$ mapping $D$ onto the plane with slits along the arcs of the logarithmic spirals that make an angle $\theta$ with rays emanating from the origin (it is a sufficient to take $- \pi / 2 \leq \theta \leq \pi / 2$; for $\theta = 0$ the logarithmic spiral degenerates into a ray emanating from the origin, while for $\theta = \pm \pi / 2$ it degenerates into a circle with centre at the origin). Let

$$p ( \zeta ) = \ \sqrt {F _ {0} ( \zeta ) F _ {\pi / 2 } ( \zeta ) } ,\ \ q ( \zeta ) = \ \sqrt { \frac{F _ {0} ( \zeta ) }{F _ {\pi / 2 } ( \zeta ) } } ,$$

where those branches of the square root are taken that give first coefficients 1 in the Laurent expansions of $p ( \zeta )$ and $q ( \zeta )$ in a neighbourhood of $\zeta = \infty$. Then the range of $\ln F ^ { \prime } ( \zeta _ {0} )$ on $\Sigma ( D)$ is the disc defined by

$$| \ln F ^ { \prime } ( \zeta _ {0} ) - \ln p ^ \prime ( \zeta _ {0} ) | \leq \ - \ln q ( \zeta _ {0} ) ,$$

where to each boundary point only the functions $F ( \zeta ) = F _ \theta ( \zeta ) + C$ with suitable $\theta$, and $C$ a constant, correspond. In particular, one has the sharp inequalities

$$| F _ {0} ^ { \prime } ( \zeta _ {0} ) | \leq \ | F ^ { \prime } ( \zeta _ {0} ) | \leq \ | F _ {\pi / 2 } ^ { \prime } ( \zeta _ {0} ) | ,$$

$$\mathop{\rm arg} F _ {\pi / 4 } ^ { \prime } ( \zeta _ {0} ) \leq \mathop{\rm arg} F ^ { \prime } ( \zeta _ {0} ) \leq \ \mathop{\rm arg} F _ {- \pi / 4 } ^ { \prime } ( \zeta _ {0} ) ,$$

#### References

 [1] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) [2] J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958) [3] V.V. Chernikov, "Extremal properties of univalent conformal mappings" , Results of investigation in mathematics and mechanics during 50 years: 1917–1967 , Tomsk (1967) pp. 23–51 (In Russian) [4] I.E. Bazilevich, , Mathematics in the USSR during 40 years: 1917–1957 , 1 , Moscow (1959) pp. 444–472 (In Russian) [5] P.P. Belinskii, "General properties of quasi-conformal mappings" , Novosibirsk (1974) (In Russian) [6] R. Kühnau, "Verzerrungssätze und Koeffizientenbedingungen vom Grunskyschen Typ für quasikonforme Abbildungen" Math. Nachrichten , 48 (1971) pp. 77–105