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''Schwarz' lemma in invariant form''
 
''Schwarz' lemma in invariant form''
  
The following generalization of the [[Schwarz lemma|Schwarz lemma]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p0727001.png" /> be a bounded regular analytic function in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p0727002.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p0727003.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p0727004.png" />. Then for any points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p0727005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p0727006.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p0727007.png" /> the non-Euclidean distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p0727008.png" /> of their images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p0727009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270010.png" /> does not exceed the non-Euclidean distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270011.png" />, i.e.
+
The following generalization of the [[Schwarz lemma|Schwarz lemma]]. Let $  w = f( z) $
 +
be a bounded regular analytic function in the unit disc $  \Omega = \{ {z \in \mathbf C } : {| z | < 1 } \} $,  
 +
$  | f( z) | \leq  1 $
 +
for $  | z | < 1 $.  
 +
Then for any points $  z _ {1} $
 +
and $  z _ {2} $
 +
in $  \Omega $
 +
the non-Euclidean distance $  d( w _ {1} , w _ {2} ) $
 +
of their images $  w _ {1} = f( z _ {1} ) $
 +
and $  w _ {2} = f( z _ {2} ) $
 +
does not exceed the non-Euclidean distance $  d( z _ {1} , z _ {2} ) $,  
 +
i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
d( w _ {1} , w _ {2} )  \leq  d( z _ {1} , z _ {2} ) ,\ \
 +
| z _ {1} | , | z _ {2} |  < 1.
 +
$$
  
 
One also has the inequality
 
One also has the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
 
 +
\frac{| dw | }{1- | w |  ^ {2} }
 +
  \leq 
 +
\frac{| dz | }{1- | z |  ^ {2} }
 +
,\ \
 +
dw  = f ^ { \prime } ( z) dz,\  | z | < 1,
 +
$$
  
between the elements of non-Euclidean length (the differential form of Pick's theorem or the Schwarz lemma). Equality applies in (1) and (2) only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270014.png" /> is a Möbius function that maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270015.png" /> onto itself (cf. [[Fractional-linear mapping|Fractional-linear mapping]]).
+
between the elements of non-Euclidean length (the differential form of Pick's theorem or the Schwarz lemma). Equality applies in (1) and (2) only if $  w = f( z) $
 +
is a Möbius function that maps $  \Omega $
 +
onto itself (cf. [[Fractional-linear mapping|Fractional-linear mapping]]).
  
The non-Euclidean, or hyperbolic, distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270016.png" /> is the distance in Lobachevskii geometry between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270018.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270019.png" /> is the Lobachevskii plane and arcs of circles serve as Lobachevskii straight lines, these being orthogonal to the unit circle (Poincaré's model), and
+
The non-Euclidean, or hyperbolic, distance $  d( z _ {1} , z _ {2} ) $
 +
is the distance in Lobachevskii geometry between $  z _ {1} $
 +
and $  z _ {2} $
 +
when $  \Omega $
 +
is the Lobachevskii plane and arcs of circles serve as Lobachevskii straight lines, these being orthogonal to the unit circle (Poincaré's model), and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270020.png" /></td> </tr></table>
+
$$
 +
d( z _ {1} , z _ {2} )  =
 +
\frac{1}{2}
 +
  \mathop{\rm ln}  ( \alpha , \beta , z _ {1} , z _ {2} ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270021.png" /></td> </tr></table>
+
$$
 +
= \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270022.png" /> is the [[Cross ratio|cross ratio]] between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270024.png" /> and the points of intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270026.png" /> of the Lobachevskii straight line passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270028.png" /> with the unit circle (see Fig.).
+
\frac{1}{2}
 +
  \mathop{\rm ln}
 +
\frac{| 1- \overline{z}\; _ {2} z _ {2} | +| z _ {1} - z _ {2} | }{| 1- \overline{z}\; _ {1} z _ {2} | - | z _ {1} - z _ {2} | }
 +
,
 +
$$
 +
 
 +
where $  ( \alpha , \beta , z _ {1} , z _ {2} ) $
 +
is the [[Cross ratio|cross ratio]] between the points $  z _ {1} $
 +
and $  z _ {2} $
 +
and the points of intersection $  \alpha $
 +
and $  \beta $
 +
of the Lobachevskii straight line passing through $  z _ {1} $
 +
and $  z _ {2} $
 +
with the unit circle (see Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072700a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072700a.gif" />
Line 23: Line 80:
 
Figure: p072700a
 
Figure: p072700a
  
The non-Euclidean length of the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270029.png" /> of any rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270030.png" /> under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270031.png" /> does not exceed the non-Euclidean length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072700/p07270032.png" />.
+
The non-Euclidean length of the image $  f( L) $
 +
of any rectifiable curve $  L \subset  \Omega $
 +
under the mapping $  w = f( z) $
 +
does not exceed the non-Euclidean length of $  L $.
  
 
The theorem was established by G. Pick [[#References|[1]]]; a far-reaching generalization of it is provided by the principle of the hyperbolic metric (cf. [[Hyperbolic metric, principle of the|Hyperbolic metric, principle of the]]). In geometric function theory these theorems provide bounds for various functionals related to mapping functions [[#References|[2]]], [[#References|[3]]].
 
The theorem was established by G. Pick [[#References|[1]]]; a far-reaching generalization of it is provided by the principle of the hyperbolic metric (cf. [[Hyperbolic metric, principle of the|Hyperbolic metric, principle of the]]). In geometric function theory these theorems provide bounds for various functionals related to mapping functions [[#References|[2]]], [[#References|[3]]].
Line 29: Line 89:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Pick,  "Ueber eine Eigenschaft der konformen Abbildung kreisförmiger Bereiche"  ''Math. Ann.'' , '''77'''  (1916)  pp. 1–6</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  C. Carathéodory,  "Conformal representation" , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.B. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Pick,  "Ueber eine Eigenschaft der konformen Abbildung kreisförmiger Bereiche"  ''Math. Ann.'' , '''77'''  (1916)  pp. 1–6</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  C. Carathéodory,  "Conformal representation" , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.B. Garnett,  "Bounded analytic functions" , Acad. Press  (1981)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Conformal invariants. Topics in geometric function theory" , McGraw-Hill  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Lang,  "Introduction to complex hyperbolic spaces" , Springer  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Conformal invariants. Topics in geometric function theory" , McGraw-Hill  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Lang,  "Introduction to complex hyperbolic spaces" , Springer  (1987)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


Schwarz' lemma in invariant form

The following generalization of the Schwarz lemma. Let $ w = f( z) $ be a bounded regular analytic function in the unit disc $ \Omega = \{ {z \in \mathbf C } : {| z | < 1 } \} $, $ | f( z) | \leq 1 $ for $ | z | < 1 $. Then for any points $ z _ {1} $ and $ z _ {2} $ in $ \Omega $ the non-Euclidean distance $ d( w _ {1} , w _ {2} ) $ of their images $ w _ {1} = f( z _ {1} ) $ and $ w _ {2} = f( z _ {2} ) $ does not exceed the non-Euclidean distance $ d( z _ {1} , z _ {2} ) $, i.e.

$$ \tag{1 } d( w _ {1} , w _ {2} ) \leq d( z _ {1} , z _ {2} ) ,\ \ | z _ {1} | , | z _ {2} | < 1. $$

One also has the inequality

$$ \tag{2 } \frac{| dw | }{1- | w | ^ {2} } \leq \frac{| dz | }{1- | z | ^ {2} } ,\ \ dw = f ^ { \prime } ( z) dz,\ | z | < 1, $$

between the elements of non-Euclidean length (the differential form of Pick's theorem or the Schwarz lemma). Equality applies in (1) and (2) only if $ w = f( z) $ is a Möbius function that maps $ \Omega $ onto itself (cf. Fractional-linear mapping).

The non-Euclidean, or hyperbolic, distance $ d( z _ {1} , z _ {2} ) $ is the distance in Lobachevskii geometry between $ z _ {1} $ and $ z _ {2} $ when $ \Omega $ is the Lobachevskii plane and arcs of circles serve as Lobachevskii straight lines, these being orthogonal to the unit circle (Poincaré's model), and

$$ d( z _ {1} , z _ {2} ) = \frac{1}{2} \mathop{\rm ln} ( \alpha , \beta , z _ {1} , z _ {2} ) = $$

$$ = \ \frac{1}{2} \mathop{\rm ln} \frac{| 1- \overline{z}\; _ {2} z _ {2} | +| z _ {1} - z _ {2} | }{| 1- \overline{z}\; _ {1} z _ {2} | - | z _ {1} - z _ {2} | } , $$

where $ ( \alpha , \beta , z _ {1} , z _ {2} ) $ is the cross ratio between the points $ z _ {1} $ and $ z _ {2} $ and the points of intersection $ \alpha $ and $ \beta $ of the Lobachevskii straight line passing through $ z _ {1} $ and $ z _ {2} $ with the unit circle (see Fig.).

Figure: p072700a

The non-Euclidean length of the image $ f( L) $ of any rectifiable curve $ L \subset \Omega $ under the mapping $ w = f( z) $ does not exceed the non-Euclidean length of $ L $.

The theorem was established by G. Pick [1]; a far-reaching generalization of it is provided by the principle of the hyperbolic metric (cf. Hyperbolic metric, principle of the). In geometric function theory these theorems provide bounds for various functionals related to mapping functions [2], [3].

References

[1] G. Pick, "Ueber eine Eigenschaft der konformen Abbildung kreisförmiger Bereiche" Math. Ann. , 77 (1916) pp. 1–6
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[3] C. Carathéodory, "Conformal representation" , Cambridge Univ. Press (1952)
[4] J.B. Garnett, "Bounded analytic functions" , Acad. Press (1981)

Comments

References

[a1] L.V. Ahlfors, "Conformal invariants. Topics in geometric function theory" , McGraw-Hill (1973)
[a2] S. Lang, "Introduction to complex hyperbolic spaces" , Springer (1987)
How to Cite This Entry:
Pick theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pick_theorem&oldid=48179
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article