View source for Distortion theorems

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''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|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

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

which determines the range of values of the functional  $   \mathop{\rm 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 |
 \mathop{\rm ln} \ 

\frac{F ( \zeta _ {1} ) - F ( \zeta _ {0} ) }{\zeta _ {1} - \zeta _ {2} }

\right |  \leq   -
 \mathop{\rm 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. [[#References|[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., [[#References|[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

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

\frac{f ( z _ {0} ) }{z _ {0} }
 ,\ \ 
 \mathop{\rm ln} \ 

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

$$

on  $  S $(
cf. [[#References|[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  $   \mathop{\rm ln}  f ( z _ {0} ) / z _ {0} $, 
$  0 < | z _ {0} | < 1 $, 
on  $  S $
is easily obtained from that of  $   \mathop{\rm 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|Schwarz lemma]] (cf. [[#References|[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  $   \mathop{\rm ln}  F ^ { \prime } ( \zeta _ {0} ) $
on  $  \Sigma ( D) $
is the disc defined by

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

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Jenkins,   "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.E. Bazilevich,  , ''Mathematics in the USSR during 40 years: 1917–1957'' , '''1''' , Moscow  (1959)  pp. 444–472  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.P. Belinskii,   "General properties of quasi-conformal mappings" , Novosibirsk  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Kühnau,   "Verzerrungssätze und Koeffizientenbedingungen vom Grunskyschen Typ für quasikonforme Abbildungen"  ''Math. Nachrichten'' , '''48'''  (1971)  pp. 77–105</TD></TR></table>

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. Chapt. 3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Pommerenke,   "Univalent functions" , Vandenhoeck &amp; Ruprecht  (1975)</TD></TR></table>