View source for Quasi-symmetric function of a complex variable

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

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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]]].

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]]].

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.

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]]].

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]]].

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]]].

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

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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]]].

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]]].

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" />,

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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" />.

====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>