Quasi-symmetric function of a complex variable
An automorphism of the real axis
(i.e. a sense-preserving homeomorphism
of
onto itself) is said to be
-quasi-symmetric on
(notation:
-
) if
![]() |
holds for all and all
. An automorphism
of
is quasi-symmetric (notation:
) if
-
for some
. A. Beurling and L.V. Ahlfors established a close relation between
and quasi-conformal mappings of the upper half-plane
onto itself (cf. also Quasi-conformal mapping), cf. statements A), B) below. The term "quasi-symmetric" was proposed in [a2].
A) Any -quasi-conformal automorphism
of
normalized by the condition
admits a homeomorphic extension to the closure of
and generates in this way
-
, where
, cf. [a1], [a6].
Here ,
, is the module of the ring domain
,
(cf. also Modulus of an annulus). The bound for
is sharp.
B) Conversely, for any there exists a constant
such that an arbitrary
-
has a quasi-conformal extension
to
with
whose maximal dilatation
satisfies
, cf. [a1], [a6].
The best value of known today (2000) is
, cf. [a5].
Quasi-symmetric functions on satisfy the following: If
, so does
; if
, so does
. However, there exist singular functions on
that are also quasi-symmetric [a1].
One may also distinguish the class -
of
-quasi-symmetric automorphisms
of the unit circle
. To this end, let
denote the length of an open arc
. Then
-
if there is an
such that for any pair
of open disjoint subarcs of
with a common end-point
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The class -
has some nice properties: no boundary point of
is distinguished, Hölder continuity is global on
and any
may be represented by an absolutely convergent Fourier series, cf. [a3], [a4].
Quasi-symmetric automorphisms of or
are intimately connected with quasi-circles, i.e. image curves of a circle under a quasi-conformal automorphism of
. Let
be a Jordan curve in the finite plane
and let
(or
) be a conformal mapping of the inside (or outside) domain of
onto
(respectively,
). Then
is an automorphism of
and
is equivalent to
being a quasi-circle [a6], [a7].
A sense-preserving homeomorphism is said to be an
-quasi-symmetric function on
(notation:
-
) if for any triple
,
,
![]() |
Obviously, -
-
. One defines
to be a quasi-symmetric function on
if
-
. For any
the Jordan curve
is a quasi-circle, cf. [a8]. The following characterization of
was given by P. Tukia and J. Väisälä in [a9]: For
with
, put
. Then
if and only if there is an automorphism
of
such that
for all admissible triples
.
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 |
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=16525