Namespaces
Variants
Actions

Difference between revisions of "Quasi-symmetric function of a complex variable"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (AUTOMATIC EDIT (latexlist): Replaced 107 formulas out of 107 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
Line 1: Line 1:
An [[Automorphism|automorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q1300501.png" /> of the real axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q1300502.png" /> (i.e. a sense-preserving [[Homeomorphism|homeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q1300503.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q1300504.png" /> onto itself) is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q1300506.png" />-quasi-symmetric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q1300507.png" /> (notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q1300508.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q1300509.png" />) if
+
<!--This article has been texified automatically. Since there was no Nroff source code for this article,
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<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/q/q130/q130050/q13005010.png" /></td> </tr></table>
+
Out of 107 formulas, 107 were replaced by TEX code.-->
  
holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005011.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005012.png" />. An automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005014.png" /> is quasi-symmetric (notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005015.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005016.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005017.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005018.png" />. A. Beurling and L.V. Ahlfors established a close relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005019.png" /> and quasi-conformal mappings of the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005020.png" /> onto itself (cf. also [[Quasi-conformal mapping|Quasi-conformal mapping]]), cf. statements A), B) below. The term  "quasi-symmetric"  was proposed in [[#References|[a2]]].
+
{{TEX|semi-auto}}{{TEX|done}}
 +
An [[Automorphism|automorphism]] $h$ of the real axis $\mathbf{R}$ (i.e. a sense-preserving [[Homeomorphism|homeomorphism]] $h$ of $\mathbf{R}$ onto itself) is said to be $M$-quasi-symmetric on $\mathbf{R}$ (notation: $h \in M$-$\operatorname {QS} ( \mathbf R )$) if
  
A) Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005021.png" />-quasi-conformal automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005023.png" /> normalized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005024.png" /> admits a homeomorphic extension to the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005025.png" /> and generates in this way <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005026.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005028.png" />, cf. [[#References|[a1]]], [[#References|[a6]]].
+
\begin{equation*} M ^ { - 1 } \leq \frac { h ( x + t ) - h ( x ) } { h ( x ) - h ( x - t ) } \leq M \end{equation*}
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005030.png" />, is the module of the ring domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005032.png" /> (cf. also [[Modulus of an annulus|Modulus of an annulus]]). The bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005033.png" /> is sharp.
+
holds for all $X \in \mathbf R$ and all $t &gt; 0$. An automorphism $h$ of $\mathbf{R}$ is quasi-symmetric (notation: $h \in \operatorname {QS} ( \mathbf{R} )$) if $h \in M$-$\operatorname {QS} ( \mathbf R )$ for some $M \geq 1$. A. Beurling and L.V. Ahlfors established a close relation between $h \in \operatorname {QS} ( \mathbf{R} )$ and quasi-conformal mappings of the upper half-plane $H$ onto itself (cf. also [[Quasi-conformal mapping|Quasi-conformal mapping]]), cf. statements A), B) below. The term  "quasi-symmetric" was proposed in [[#References|[a2]]].
  
B) Conversely, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005034.png" /> there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005035.png" /> such that an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005036.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005037.png" /> has a quasi-conformal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005038.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005039.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005040.png" /> whose maximal dilatation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005041.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005042.png" />, cf. [[#References|[a1]]], [[#References|[a6]]].
+
A) Any $K$-quasi-conformal automorphism $f$ of $H$ normalized by the condition $f ( \infty ) = \infty$ admits a homeomorphic extension to the closure of $H$ and generates in this way $h \in M$-$\operatorname {QS} ( \mathbf R )$, where $M = \lambda ( K ) : = [ \mu ^ { - 1 } ( \pi K / 2 ) ] ^ { - 2 } - 1$, cf. [[#References|[a1]]], [[#References|[a6]]].
  
The best value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005043.png" /> known today (2000) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005044.png" />, cf. [[#References|[a5]]].
+
Here $\mu ( r )$, $0 &lt; r &lt; 1$, is the module of the ring domain $\mathbf{D} \backslash [ 0 , r ]$, $\mathbf{D} = \{ z \in \mathbf{C} : | z | &lt; 1 \}$ (cf. also [[Modulus of an annulus|Modulus of an annulus]]). The bound for $M$ is sharp.
  
Quasi-symmetric functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005045.png" /> satisfy the following: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005046.png" />, so does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005047.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005048.png" />, so does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005049.png" />. However, there exist singular functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005050.png" /> that are also quasi-symmetric [[#References|[a1]]].
+
B) Conversely, for any $M \geq 1$ there exists a constant $K ( M )$ such that an arbitrary $h \in M$-$\operatorname {QS} ( \mathbf R )$ has a quasi-conformal extension $f$ to $H$ with $f ( \infty ) = \infty$ whose maximal dilatation $K [ f ]$ satisfies $K [ f ] \leq K ( M )$, cf. [[#References|[a1]]], [[#References|[a6]]].
  
One may also distinguish the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005051.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005052.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005053.png" />-quasi-symmetric automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005054.png" /> of the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005055.png" />. To this end, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005056.png" /> denote the length of an open arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005057.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005058.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005059.png" /> if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005060.png" /> such that for any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005061.png" /> of open disjoint subarcs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005062.png" /> with a common end-point
+
The best value of $K ( M )$ known today (2000) is $\operatorname{min} \{ M ^ { 3 / 2 } , 2 M - 1 \}$, cf. [[#References|[a5]]].
  
<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/q/q130/q130050/q13005063.png" /></td> </tr></table>
+
Quasi-symmetric functions on $\mathbf{R}$ satisfy the following: If $h \in \operatorname {QS} ( \mathbf{R} )$, so does $h ^ { - 1 }$; if $h _ { 1 } , h _ { 2 } \in \operatorname {QS} ( \mathbf{R} )$, so does $h _ { 1 } \circ h _ { 2 }$. However, there exist singular functions on $\mathbf{R}$ that are also quasi-symmetric [[#References|[a1]]].
  
The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005064.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005065.png" /> has some nice properties: no boundary point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005066.png" /> is distinguished, Hölder continuity is global on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005067.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005068.png" /> may be represented by an absolutely convergent [[Fourier series|Fourier series]], cf. [[#References|[a3]]], [[#References|[a4]]].
+
One may also distinguish the class $M$-$\operatorname{QS} ( \mathbf{T} )$ of $M$-quasi-symmetric automorphisms $h$ of the unit circle $\mathbf{T} = \partial \mathbf D $. To this end, let $| \alpha |$ denote the length of an open arc $\alpha \subset \mathbf{T}$. Then $h \in M$-$\operatorname{QS} ( \mathbf{T} )$ if there is an $M \geq 1$ such that for any pair $\alpha , \beta$ of open disjoint subarcs of $\bf T$ with a common end-point
  
Quasi-symmetric automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005069.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005070.png" /> are intimately connected with quasi-circles, i.e. image curves of a circle under a quasi-conformal automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005071.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005072.png" /> be a [[Jordan curve|Jordan curve]] in the finite plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005073.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005074.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005075.png" />) be a [[Conformal mapping|conformal mapping]] of the inside (or outside) domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005076.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005077.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005078.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005079.png" /> is an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005081.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005082.png" /> being a quasi-circle [[#References|[a6]]], [[#References|[a7]]].
+
\begin{equation*} | \alpha | = | \beta | \Rightarrow \frac { | h ( \alpha ) | } { | h ( \beta ) | } \leq M. \end{equation*}
  
A sense-preserving homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005083.png" /> is said to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005086.png" />-quasi-symmetric function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005087.png" /> (notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005088.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005089.png" />) if for any triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005091.png" />,
+
The class $\operatorname{QS} ( \mathbf{T} ) = \cup _ { M \geq 1 } M$-$\operatorname{QS} ( \mathbf{T} )$ has some nice properties: no boundary point of $\mathbf D$ is distinguished, Hölder continuity is global on $\bf T$ and any $h \in \operatorname { QS} ( \mathbf{T} )$ may be represented by an absolutely convergent [[Fourier series|Fourier series]], cf. [[#References|[a3]]], [[#References|[a4]]].
  
<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/q/q130/q130050/q13005092.png" /></td> </tr></table>
+
Quasi-symmetric automorphisms of $\mathbf{R}$ or $\bf T$ are intimately connected with quasi-circles, i.e. image curves of a circle under a quasi-conformal automorphism of $\widehat{\mathbf{C}}$. Let $\mathcal{J}$ be a [[Jordan curve|Jordan curve]] in the finite plane $\mathbf{C}$ and let $f$ (or $F$) be a [[Conformal mapping|conformal mapping]] of the inside (or outside) domain of $\mathcal{J}$ onto $\mathbf D$ (respectively, $\mathbf{D} ^ { * } = \widehat { \mathbf{C} } \backslash \overline { \mathbf{D} }$). Then $h = F \circ f ^ { - 1 }$ is an automorphism of $\bf T$ and $h \in \operatorname { QS} ( \mathbf{T} )$ is equivalent to $\mathcal{J}$ being a quasi-circle [[#References|[a6]]], [[#References|[a7]]].
  
Obviously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005093.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005094.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005095.png" />. One defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005096.png" /> to be a quasi-symmetric function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005098.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q13005099.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050100.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050101.png" /> the Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050102.png" /> is a quasi-circle, cf. [[#References|[a8]]]. The following characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050103.png" /> was given by P. Tukia and J. Väisälä in [[#References|[a9]]]: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050104.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050105.png" />, put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050106.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050107.png" /> if and only if there is an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050108.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050109.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050110.png" /> for all admissible triples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q130/q130050/q130050111.png" />.
+
A sense-preserving homeomorphism $h : \mathbf{T} \rightarrow \mathbf{C}$ is said to be an $M$-quasi-symmetric function on $\bf T$ (notation: $h \in M$-$\operatorname{QS} ( \mathbf{T} , \mathbf{C} )$) if for any triple $z _ { 1 } , z _ { 2 } , z _ { 3 } \in \mathbf{T}$, $z _ { 2 } \neq z _ { 3 }$,
 +
 
 +
\begin{equation*} | z _ { 1 } - z _ { 2 } | = | z _ { 2 } - z _ { 3 } | \Rightarrow \frac { | h ( z _ { 1 } ) - h ( z _ { 2 } ) | } { | h ( z _ { 2 } ) - h ( z _ { 3 } ) | } \leq M. \end{equation*}
 +
 
 +
Obviously, $M$-$\operatorname{QS} ( {\bf T} ) \subset M$-$\operatorname{QS} ( \mathbf{T} , \mathbf{C} )$. One defines $h$ to be a quasi-symmetric function on $\bf T$ if $h \in \operatorname{QS} (\mathbf{ T} , \mathbf{C} ) : = \cup _ { M \geq 1 } M$-$\operatorname{QS} ( \mathbf{T} , \mathbf{C} )$. For any $h \in \operatorname{QS} ( \mathbf{T} , \mathbf{C} )$ the Jordan curve $h ( \mathbf{T} )$ is a quasi-circle, cf. [[#References|[a8]]]. The following characterization of $\operatorname{QS} ( \mathbf{T} , \mathbf{C} )$ was given by P. Tukia and J. Väisälä in [[#References|[a9]]]: For $a , b , x \in \mathbf{T}$ with $b \neq x$, put $\rho = | a - x | / | b - x |$. Then $h \in \operatorname{QS} ( \mathbf{T} , \mathbf{C} )$ if and only if there is an automorphism $ \eta $ of $[ 0 , + \infty )$ such that $| h ( a ) - h ( x ) | / | h ( b ) - h ( x ) | \leq \eta ( \rho )$ for all admissible triples $a , b , x$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Beurling,  L.V. Ahlfors,  "The boundary correspondence under quasiconformal mappings"  ''Acta Math.'' , '''96'''  (1956)  pp. 125–142</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Kelingos,  "Contributions to the theory of quasiconformal mappings" , Diss. Univ. Michigan  (1963)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.G. Krzyż,  "Quasicircles and harmonic measure"  ''Ann. Acad. Sci. Fenn. Ser. A.I. Math.'' , '''12'''  (1987)  pp. 19–24</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.G. Krzyż,  M. Nowak,  "Harmonic automorphisms of the unit disk"  ''J. Comput. Appl. Math.'' , '''105'''  (1999)  pp. 337–346</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Lehtinen,  "Remarks on the maximal dilatations of the Beurling–Ahlfors extension"  ''Ann. Acad. Sci. Fenn. Ser. A.I. Math.'' , '''9'''  (1984)  pp. 133–139</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  O. Lehto,  K.I. Virtanen,  "Quasiconformal mappings in the plane" , Springer  (1973)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Partyka,  "A sewing theorem for complementary Jordan domains"  ''Ann. Univ. Mariae Curie–Skłodowska Sect. A'' , '''41'''  (1987)  pp. 99–103</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  Ch. Pommerenke,  "Boundary behaviour of conformal maps" , Springer  (1992)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  P. Tukia,  J. Väisälä,  "Quasisymmetric embeddings of metric spaces"  ''Ann. Acad. Sci. Fenn. Ser. A.I. Math.'' , '''5'''  (1980)  pp. 97–114</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  A. Beurling,  L.V. Ahlfors,  "The boundary correspondence under quasiconformal mappings"  ''Acta Math.'' , '''96'''  (1956)  pp. 125–142</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.A. Kelingos,  "Contributions to the theory of quasiconformal mappings" , Diss. Univ. Michigan  (1963)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J.G. Krzyż,  "Quasicircles and harmonic measure"  ''Ann. Acad. Sci. Fenn. Ser. A.I. Math.'' , '''12'''  (1987)  pp. 19–24</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J.G. Krzyż,  M. Nowak,  "Harmonic automorphisms of the unit disk"  ''J. Comput. Appl. Math.'' , '''105'''  (1999)  pp. 337–346</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M. Lehtinen,  "Remarks on the maximal dilatations of the Beurling–Ahlfors extension"  ''Ann. Acad. Sci. Fenn. Ser. A.I. Math.'' , '''9'''  (1984)  pp. 133–139</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  O. Lehto,  K.I. Virtanen,  "Quasiconformal mappings in the plane" , Springer  (1973)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  D. Partyka,  "A sewing theorem for complementary Jordan domains"  ''Ann. Univ. Mariae Curie–Skłodowska Sect. A'' , '''41'''  (1987)  pp. 99–103</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  Ch. Pommerenke,  "Boundary behaviour of conformal maps" , Springer  (1992)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  P. Tukia,  J. Väisälä,  "Quasisymmetric embeddings of metric spaces"  ''Ann. Acad. Sci. Fenn. Ser. A.I. Math.'' , '''5'''  (1980)  pp. 97–114</td></tr></table>

Revision as of 16:45, 1 July 2020

An automorphism $h$ of the real axis $\mathbf{R}$ (i.e. a sense-preserving homeomorphism $h$ of $\mathbf{R}$ onto itself) is said to be $M$-quasi-symmetric on $\mathbf{R}$ (notation: $h \in M$-$\operatorname {QS} ( \mathbf R )$) if

\begin{equation*} M ^ { - 1 } \leq \frac { h ( x + t ) - h ( x ) } { h ( x ) - h ( x - t ) } \leq M \end{equation*}

holds for all $X \in \mathbf R$ and all $t > 0$. An automorphism $h$ of $\mathbf{R}$ is quasi-symmetric (notation: $h \in \operatorname {QS} ( \mathbf{R} )$) if $h \in M$-$\operatorname {QS} ( \mathbf R )$ for some $M \geq 1$. A. Beurling and L.V. Ahlfors established a close relation between $h \in \operatorname {QS} ( \mathbf{R} )$ and quasi-conformal mappings of the upper half-plane $H$ onto itself (cf. also Quasi-conformal mapping), cf. statements A), B) below. The term "quasi-symmetric" was proposed in [a2].

A) Any $K$-quasi-conformal automorphism $f$ of $H$ normalized by the condition $f ( \infty ) = \infty$ admits a homeomorphic extension to the closure of $H$ and generates in this way $h \in M$-$\operatorname {QS} ( \mathbf R )$, where $M = \lambda ( K ) : = [ \mu ^ { - 1 } ( \pi K / 2 ) ] ^ { - 2 } - 1$, cf. [a1], [a6].

Here $\mu ( r )$, $0 < r < 1$, is the module of the ring domain $\mathbf{D} \backslash [ 0 , r ]$, $\mathbf{D} = \{ z \in \mathbf{C} : | z | < 1 \}$ (cf. also Modulus of an annulus). The bound for $M$ is sharp.

B) Conversely, for any $M \geq 1$ there exists a constant $K ( M )$ such that an arbitrary $h \in M$-$\operatorname {QS} ( \mathbf R )$ has a quasi-conformal extension $f$ to $H$ with $f ( \infty ) = \infty$ whose maximal dilatation $K [ f ]$ satisfies $K [ f ] \leq K ( M )$, cf. [a1], [a6].

The best value of $K ( M )$ known today (2000) is $\operatorname{min} \{ M ^ { 3 / 2 } , 2 M - 1 \}$, cf. [a5].

Quasi-symmetric functions on $\mathbf{R}$ satisfy the following: If $h \in \operatorname {QS} ( \mathbf{R} )$, so does $h ^ { - 1 }$; if $h _ { 1 } , h _ { 2 } \in \operatorname {QS} ( \mathbf{R} )$, so does $h _ { 1 } \circ h _ { 2 }$. However, there exist singular functions on $\mathbf{R}$ that are also quasi-symmetric [a1].

One may also distinguish the class $M$-$\operatorname{QS} ( \mathbf{T} )$ of $M$-quasi-symmetric automorphisms $h$ of the unit circle $\mathbf{T} = \partial \mathbf D $. To this end, let $| \alpha |$ denote the length of an open arc $\alpha \subset \mathbf{T}$. Then $h \in M$-$\operatorname{QS} ( \mathbf{T} )$ if there is an $M \geq 1$ such that for any pair $\alpha , \beta$ of open disjoint subarcs of $\bf T$ with a common end-point

\begin{equation*} | \alpha | = | \beta | \Rightarrow \frac { | h ( \alpha ) | } { | h ( \beta ) | } \leq M. \end{equation*}

The class $\operatorname{QS} ( \mathbf{T} ) = \cup _ { M \geq 1 } M$-$\operatorname{QS} ( \mathbf{T} )$ has some nice properties: no boundary point of $\mathbf D$ is distinguished, Hölder continuity is global on $\bf T$ and any $h \in \operatorname { QS} ( \mathbf{T} )$ may be represented by an absolutely convergent Fourier series, cf. [a3], [a4].

Quasi-symmetric automorphisms of $\mathbf{R}$ or $\bf T$ are intimately connected with quasi-circles, i.e. image curves of a circle under a quasi-conformal automorphism of $\widehat{\mathbf{C}}$. Let $\mathcal{J}$ be a Jordan curve in the finite plane $\mathbf{C}$ and let $f$ (or $F$) be a conformal mapping of the inside (or outside) domain of $\mathcal{J}$ onto $\mathbf D$ (respectively, $\mathbf{D} ^ { * } = \widehat { \mathbf{C} } \backslash \overline { \mathbf{D} }$). Then $h = F \circ f ^ { - 1 }$ is an automorphism of $\bf T$ and $h \in \operatorname { QS} ( \mathbf{T} )$ is equivalent to $\mathcal{J}$ being a quasi-circle [a6], [a7].

A sense-preserving homeomorphism $h : \mathbf{T} \rightarrow \mathbf{C}$ is said to be an $M$-quasi-symmetric function on $\bf T$ (notation: $h \in M$-$\operatorname{QS} ( \mathbf{T} , \mathbf{C} )$) if for any triple $z _ { 1 } , z _ { 2 } , z _ { 3 } \in \mathbf{T}$, $z _ { 2 } \neq z _ { 3 }$,

\begin{equation*} | z _ { 1 } - z _ { 2 } | = | z _ { 2 } - z _ { 3 } | \Rightarrow \frac { | h ( z _ { 1 } ) - h ( z _ { 2 } ) | } { | h ( z _ { 2 } ) - h ( z _ { 3 } ) | } \leq M. \end{equation*}

Obviously, $M$-$\operatorname{QS} ( {\bf T} ) \subset M$-$\operatorname{QS} ( \mathbf{T} , \mathbf{C} )$. One defines $h$ to be a quasi-symmetric function on $\bf T$ if $h \in \operatorname{QS} (\mathbf{ T} , \mathbf{C} ) : = \cup _ { M \geq 1 } M$-$\operatorname{QS} ( \mathbf{T} , \mathbf{C} )$. For any $h \in \operatorname{QS} ( \mathbf{T} , \mathbf{C} )$ the Jordan curve $h ( \mathbf{T} )$ is a quasi-circle, cf. [a8]. The following characterization of $\operatorname{QS} ( \mathbf{T} , \mathbf{C} )$ was given by P. Tukia and J. Väisälä in [a9]: For $a , b , x \in \mathbf{T}$ with $b \neq x$, put $\rho = | a - x | / | b - x |$. Then $h \in \operatorname{QS} ( \mathbf{T} , \mathbf{C} )$ if and only if there is an automorphism $ \eta $ of $[ 0 , + \infty )$ such that $| h ( a ) - h ( x ) | / | h ( b ) - h ( x ) | \leq \eta ( \rho )$ for all admissible triples $a , b , x$.

References

[a1] A. Beurling, L.V. Ahlfors, "The boundary correspondence under quasiconformal mappings" Acta Math. , 96 (1956) pp. 125–142
[a2] J.A. Kelingos, "Contributions to the theory of quasiconformal mappings" , Diss. Univ. Michigan (1963)
[a3] J.G. Krzyż, "Quasicircles and harmonic measure" Ann. Acad. Sci. Fenn. Ser. A.I. Math. , 12 (1987) pp. 19–24
[a4] J.G. Krzyż, M. Nowak, "Harmonic automorphisms of the unit disk" J. Comput. Appl. Math. , 105 (1999) pp. 337–346
[a5] M. Lehtinen, "Remarks on the maximal dilatations of the Beurling–Ahlfors extension" Ann. Acad. Sci. Fenn. Ser. A.I. Math. , 9 (1984) pp. 133–139
[a6] O. Lehto, K.I. Virtanen, "Quasiconformal mappings in the plane" , Springer (1973)
[a7] D. Partyka, "A sewing theorem for complementary Jordan domains" Ann. Univ. Mariae Curie–Skłodowska Sect. A , 41 (1987) pp. 99–103
[a8] Ch. Pommerenke, "Boundary behaviour of conformal maps" , Springer (1992)
[a9] P. Tukia, J. Väisälä, "Quasisymmetric embeddings of metric spaces" Ann. Acad. Sci. Fenn. Ser. A.I. Math. , 5 (1980) pp. 97–114
How to Cite This Entry:
Quasi-symmetric function of a complex variable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-symmetric_function_of_a_complex_variable&oldid=49952
This article was adapted from an original article by Jan G. Krzyż (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article