Quasi-symmetric function of a complex variable

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