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

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

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

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Other distortion theorems are, e.g., Landau's theorems (cf. Landau theorems), Bloch's theorem (cf. Bloch constant) and the Pick theorem.

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