Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 167–178

ON BOUNDARY CORRESPONDENCE UNDER QUASICONFORMAL MAPPINGS

V.Ya. Gutlyanskiǐ and V.I. Ryazanov

Ukrainian Academy of Sciences, Institute for Applied Mathematics and Mechanics Roze Luxemburg str. 74, 340114, Donetsk, Ukraine

Abstract. We study boundary properties of quasiconformal self-mappings depending on complex dilatations. We give some new conditions for the corresponding quasisymmetric function to be asymptotically symmetric and obtain an explicit asymptotical representation for the distortion ratio of boundary correspondence when the complex dilatation has directional limits.

1. Introduction

Let \overline{C} be the extended complex plane. We shall denote the unit disk |z| < 1 by D, the unit sphere |z| = 1 by S^{1} and the upper half-plane Im z > 0 by H.

An orientation preserving 𝒜 𝒞 ℒ homeomorphism f is called quasiconformal if it satisfies the Beltrami equation

(1.1)  fz̄ = μ(z)f_{z}  a.e.

for some measurable complex function μ called the Beltrami differential or complex dilatation with

(1.2)  ess sup_{G} |μ(z)| = ∥μ∥_{∞} ≤ k < 1.

If Q = (1 + k)/(1 − k) then f is called also Q-quasiconformal.

An increasing self-homeomorphism f of the real axis R is called quasisymmetric if it can be extended to a quasiconformal mapping of the upper half-plane H that fixes the point at infinity.

It is a well-known fundamental result that the boundary values of a Qquasiconformal self-mapping f of the upper half-plane preserving the point at infinity satisfy the double sharp inequality (see [20, p. 81])

(1.3)  \frac{1}{λ(Q)} ≤ \frac{f(x + t) − f(x)}{f(x) − f(x − t)} ≤ λ(Q)

1991 Mathematics Subject Classification: Primary 30C55, 30C60.

for all real x and t ≠ 0 with

λ(Q) = \frac{1}{(m^{−1}(πQ/2))^{2}} − 1

and m(r) from Grötzsch’s module theorem.

In 1956 Beurling and Ahlfors [4] proved the converse having shown that f is quasisymmetric if for some constant M, 1 ≤ M < ∞, the M -condition

(1.4)  \frac{1}{M} ≤ \frac{f(x + t) − f(x)}{f(x) − f(x − t)} ≤ M

is satisfied for every symmetric triple x − t, x and x + t in R.

Next in 1967 Carleson [6] considered quasiconformal self-mappings of H which “are conformal at the boundary” in the sense that

(1.5)  k(t) = ess sup_{0<Im z≤t} |μ(z)| → 0,  t → 0

and showed that the assumption

(1.6)  ∫_{0} k(t)^{α} \frac{dt}{t} < ∞

for α = 2 implies an absolutely continuous boundary correspondence f(x), while for α = 1, f(x) is continuously differentiable. He gave also counterexamples in the case when (1.6) does not hold for α = 1 or α = 2, respectively. Then Anderson, Becker and Lesley [2] investigated the same problems for quasiconformal self-mappings of the disk D by methods of the theory of conformal mappings with asymptotically conformal extension [3]. They have obtained also estimations of the moduli of continuity of f′(e^{it}) for α = 1 and estimations of the mean oscillation of ln |f′(e^{it})| ∈ VMO(S^{1}) for α = 2.

This paper is concerned with an investigation of symmetry and regularity problems at a given point and on subsets of the complex plane for quasiconformal mappings. It contains some new results related to asymptotical behavior of quasisymmetric homeomorphisms. It is well-known by a principal result of Gehring and Lehto [13] that a general quasiconformal mapping f, being Hölder continuous, is differentiable only almost everywhere and it is necessary to require some additional restrictions on the complex dilatation μ(z) in order to make more precise statements about the pointwise behaviour of the mapping. We shall choose the upper half-plane as a main canonical domain and the limit

(1.7)  lim_{t→0} \frac{1}{t^{2}} ∫∫_{|z−x|≤t,z∈H} |μ(z) − ν_{x}(z)|^{α} dm_{z} = 0,  α > 0

as a condition for measuring the deviation for the complex dilatation μ(z). Here ν_{x}(z), x ∈ ℳ ⊆ R is an appropriate one-parameter family of the Beltrami differentials. The assumptions we shall require are motivated by problems which arise in the theory of convergence and compactness for quasiconformal mappings.

We first require (1.7) to hold for ν_{x}(z) = ν_{x}, x ∈ R, that is, ν_{x}(z) does not depend on z and prove a result which implies a local asymptotical symmetry at the point x for the corresponding quasisymmetric function

(1.8)  lim_{t→0} \frac{f(x + t) − f(x)}{f(x) − f(x − t)} = 1.

The condition we impose on μ(z) does not imply the differentiability of f at the point x even if the complex dilatation is continuous at this point.

Then we focus our attention to the study of a boundary correspondence under the similar, but slightly less restrictive, requirement on μ(z) that (1.7) holds for a function ν_{x}(z) depending on arg(z − x) only for any fixed x ∈ ℳ ⊆ R. In this case we prove the more general Theorem 2 about regularity and symmetry under quasiconformal mappings. Note that a complex dilatation μ(z) having for fixed x ∈ R and almost all θ, 0 ≤ θ ≤ π, finite limits in directions

(1.9)  lim_{t→0} μ(x + te^{iθ}) = ν_{x}(e^{iθ})

satisfies the above restriction.

Finally, we apply special cases of the previous results to the study of the boundary behaviour of quasiconformal self-homeomorphisms of the unit disk D.

Different analytic and geometric properties of quasisymmetric homeomorphisms have been investigated, for instance, by Gardiner and Sullivan [9], Gehring [10], Douady and Earle [7], Fehlmann [8], Hayman [17] and Tukia [22]. Note also that quasisymmetric homeomorphisms of an interval into R play an important role in the theory of Riemann surfaces and of real one dimensional smooth dynamical systems.

2. Main results

We begin with a result on asymptotical symmetry of quasisymmetric selfhomeomorphisms of the real axis R.

Theorem 1. Let f be a Q-quasiconformal self-mapping of the upper halfplane H with f(∞) = ∞ and complex dilatation μ. If there exists a complex number ν_{x} such that

(2.1)  lim_{t→0} \frac{1}{t^{2}} ∫∫_{|z−x|≤t,z∈H} |μ(z) − ν_{x}|^{α} dm_{z} = 0

for some α > 0 and fixed x ∈ R, then

(2.2)  lim_{{t→0 \atop t∈R\\{0\}}} \frac{f(x + t) − f(x)}{f(x) − f(x − t)} = 1.

Moreover, if the limit (2.1) is uniform with respect to the parameter x ∈ ℳ ⊂ R then (2.2) is also uniform in ℳ .

In what follows 𝔉_{Q} is the class of all Q-quasiconformal self-mappings f of the extended complex plane \overline{C} normalized with the conditions f(0) = 0, f(1) = 1, f(∞) = ∞.

Proof. Suppose that f is a Q-quasiconformal mapping of H onto itself, normalized so that f(∞) = ∞. Then we can extend f by reflection in the halfplane Im z < 0 to obtain a Q-quasiconformal mapping of the complex plane \overline{C} with f(∞) = ∞. The complex dilatation μ(z) = fz̄/f_{z} will satisfy the symmetry condition μ(z̄) = \overline{μ(z)} almost everywhere in C.

Next, fix x ∈ R and introduce the family of quasiconformal mappings Φ(z, x, t) = g(tz; x)/g(t; x), where g(z; x) = f(x+z)−f(x). It is easy to verify that μ(x+tz) represents the complex dilatation for the mapping Φ(z, x, t) ∈ 𝔉_{Q}. On the other hand, the assumption (2.1) implies that μ(x+tz) ⟶^{mes} ν_{x} as t → 0, t > 0, z ∈ H. It is well-known that the last assertion implies the existence of a sequence μ(x+t_{n}z) which converges to ν_{x} almost everywhere.

Now we shall show that

(2.3)  lim_{t→0} Φ(z, x, t) = ω_{x}(z)

for all z ∈ C, where

ω_{x}(z) = \{ \begin{matrix}\frac{z + ν_{x}z̄}{1 + ν_{x}} & for Im z ≥ 0, \\ \frac{z + \overline{ν_{x}}z̄}{1 + \overline{ν_{x}}} & for Im z ≤ 0. \end{matrix}

Suppose that (2.3) fails. Then there exists an ε > 0 and a sequence t_{n} → 0 as n → ∞ such that for some z_{0} ∈ C

|Φ(z_{0}, x, t_{n}) − ω_{x}(z_{0})| ≥ ε.

Without loss of generality we can assume also that the corresponding sequence of quasiconformal mappings Φ(z, x, t_{n}) ∈ 𝔉_{Q} converges locally uniformly to some function Φ(z, x) ∈ 𝔉_{Q} and μ(x+t_{n}z), z ∈ H, converges to ν_{x} almost everywhere. Now by the Bers–Bojarski convergence theorem (see [20, p. 187]) the complex dilatation of the limit function Φ(z, x) ∈ 𝔉_{Q} will agree with ν_{x} for almost all z ∈ \{z : Im z ≥ 0\} and, by symmetry, with \overline{ν_{x}} for almost all z ∈ \{z : Im z ≤ 0\}. Hence Φ(z, x) = ω_{x}(z). This contradicts the above assumption.

Setting z = −1 in (2.3) we complete the proof of the local variant of Theorem 1 emphasizing by the same token the role of fundamental convergence theorems in our considerations.

It is easy to see that Theorem 1 is the partial case of a more general Theorem 2 below. However we decided to extract it as an independent statement to call attention to the symmetry notion for quasisymmetric functions f: R → R, being expressed by the uniform condition (2.2) in connection with new remarkable applications of the concept given by Gardiner and Sullivan [9].

Going over to the analysis of the above result we first note that if f is differentiable at x ∈ ℳ then evidently (2.2) holds. On the other hand, the following example

f(z) = z(1 − log |z|), f(0) = 0

shows that even the continuity of the complex dilatation in a neighbourhood of a prescribed point does not imply a differentiability of the corresponding quasiconformal mapping at this point. In connection with it the following statement may have independent interest.

Corollary 1. Let f be a quasiconformal self-mapping of the upper half-plane H with complex dilatation μ that can be extended to a function being continuous uniformly in ℳ ⊂ R. Then (2.2) holds uniformly in ℳ .

Indeed, the convergence of the complex dilatation μ(z) to μ(x) = ν_{x} as z → x uniformly with respect to x ∈ ℳ implies that the limit (2.1) is also uniform in ℳ for any α > 0.

Denote by δ(t, x) the essential module of continuity for a complex dilatation μ(z) extended by μ(x) at the prescribed point x ∈ R that is,

(2.4)  δ(t, x) = ess sup_{{|z−x|≤t \atop Im z>0}} |μ(z) − μ(x)|.

Corollary 2. If for some α > 0 the Dini condition

(2.5)  lim_{ε→0} ∫_{0}^{ε} \frac{δ^{α}(t, x)}{t} dt = 0

holds uniformly with respect to x ∈ ℳ then (2.2) holds also uniformly in x ∈ ℳ .

The proof of Corollary 2 follows immediately from the evident inequalities

(2.6)  \begin{aligned}\frac{1}{t^{2}} ∫∫_{|z−x|≤t,z∈H} |μ(z) − μ(x)|^{α} dm_{z} & ≤ ∫∫_{|z−x|≤t,z∈C} \frac{|μ(z) − μ(x)|^{α}}{|z − x|^{2}} dm_{z} \\ & ≤ 2π ∫_{0} \frac{δ^{α}(t, x)}{t} dt. \end{aligned}

Note that the intermediate integral has been introduced by Teichmüller, Wittich, Belinskii and Lehto (see [20, p. 232]) to prove that its convergence for α ≤ 1 implies the differentiability of f at the point x. It can be shown that the uniform convergence of the integral on ℳ ⊂ R for α ≤ 1 implies the uniform differentiability of f in ℳ . Moreover, assumption (2.1) does not imply in general the Teichmüller–Wittich–Belinskii–Lehto condition and the last one does not imply the Dini condition (see [15], [16]).

Before giving the following principal statements we need two preliminary results. The first one deals with an explicit representation of normalized quasiconformal self-mappings of the complex plane C for a special class of complex dilatations which is due to Schatz [21].

Proposition 1. Let ω be a quasiconformal self-mapping of the complex plane normalized by the conditions ω(0) = 0, ω(1) = 1, ω(∞) = ∞ with complex dilatation ν depending on arg z only. Then

(2.7)  ω(z) = \{|z| exp(i ∫_{0}^{arg z} \frac{e^{2iθ} − ν(e^{iθ})}{e^{2iθ} + ν(e^{iθ})} dθ)\}^{1/b},

where

(2.8)  b = \frac{1}{2π} ∫_{0}^{2π} \frac{e^{2iθ} − ν(e^{iθ})}{e^{2iθ} + ν(e^{iθ})} dθ.

The proof follows by straightforward verification.

Remark. The radial lines can be transformed by ω to spirals if and only if Im b ≠ 0. If Im b = 0 then all radial lines are translated to radial lines. Moreover, the real axis and upper half-plane are preserved if and only if

(2.9)  Re ∫_{0}^{π} \frac{e^{2iθ} − ν(e^{iθ})}{e^{2iθ} + ν(e^{iθ})} dθ = Re ∫_{π}^{2π} \frac{e^{2iθ} − ν(e^{iθ})}{e^{2iθ} + ν(e^{iθ})} dθ.
Note that ν(e^{−iθ}) = \overline{ν(e^{iθ})} implies (2.9) and therefore b = Re c, where
(2.10)  c = \frac{1}{π} ∫_{0}^{π} \frac{e^{2iθ} − ν(e^{iθ})}{e^{2iθ} + ν(e^{iθ})} dθ.
Example. If ν = κ(z/z̄), κ ∈ C, |κ| < 1, then ω is given by
(2.11)  ω(z) = \frac{z}{|z|}|z|^{c_{0}},

where

(2.12)  c_{0} = \frac{1 + κ}{1 − κ}

is the corresponding complex parameter with positive real part.

If we set c_{0} = α − i, α > 0, we obtain a quasiconformal mapping of C

(2.13)  h(re^{iθ}) = r^{α}e^{i(θ−ln r)}

mapping the lines arg z = const onto spirals. Note that such mappings have been used by Gehring [12] to solve the Bers problem about the structure of the universal Teichmüller space.

The following lemma connects some convergence and regularity problems for quasiconformal mappings and may have an independent interest.

Let j ∈ J be an abstract parameter. Denote by ω_{j}(z) quasiconformal mappings (2.7) with complex dilatations ν_{j} depending on arg z only.

Lemma. Let f_{j}: C → C, f_{j}(0) = 0, j ∈ J, be a family of Q-quasiconformal mappings with complex dilatations μ_{j} such that for some α > 0

(2.14)  lim_{t→0} \frac{1}{t^{2}} ∫∫_{|z|≤t} |μ_{j}(z) − ν_{j}(z)|^{α} dm_{z} = 0

uniformly with respect to j ∈ J. Then

(2.15)  lim_{{t→0 \atop t∈R\\{0\}}} \frac{f_{j}(tz)}{f_{j}(t)} = ω_{j}(z),  for all z ∈ C,

uniformly in j ∈ J.

The proof of the lemma will be a main objective of Section 3.

Theorem 2. Let f be a Q-quasiconformal self-mapping of the upper halfplane H with f(∞) = ∞ and complex dilatation μ. If there exists a complexvalued function ν_{x}(η), x ∈ ℳ ⊆ R, η ∈ S^{1} , such that for some α > 0

(2.16)  lim_{t→0} \frac{1}{t^{2}} ∫∫_{|z−x|≤t,z∈H} |μ(z) − ν_{x}(e^{iθ})|^{α} dm_{z} = 0

where θ = arg(z − x), uniformly with respect to x ∈ ℳ , then

(2.17)  lim_{{t→0 \atop t∈R\\{0\}}} \frac{f(x + τt) − f(x)}{f(x + t) − f(x)} = ω_{x}(τ),  for all τ ∈ R

uniformly with respect to x ∈ ℳ . Here ω_{x}(z) is given by (2.7) with

ν(e^{iθ}) = \{ \begin{matrix}ν_{x}(e^{iθ}) & for 0 < θ < π, \\ \overline{ν_{x}(e^{−iθ})} & for −π < θ < 0. \end{matrix}

Proof. Suppose that f is a Q-quasiconformal mapping of H onto itself normalized so that f(∞) = ∞. We shall assume f to be extended in the complex plane by reflection.

Now set J = ℳ ⊆ R, j = x ∈ ℳ , f_{j}(z) = f(x + z) − f(x), ω_{j}(z) = ω_{x}(z). Applying the lemma to the family f_{j}(z) yields the theorem.

Corollary 3. Under the hypothesis of Theorem 2

(2.18)  lim_{{t→0 \atop t>0}} \frac{f(x + t) − f(x)}{f(x) − f(x − t)} = e^{π Im c/ Re c},

(2.19)  lim_{{t→0 \atop t∈R\\{0\}}} \frac{f(x + τt) − f(x)}{f(x + t) − f(x)} = τ^{1/ Re c},  for all τ > 0

uniformly with respect to x ∈ ℳ ⊆ R.

Let ω_{x}(z) is defined by (2.7). Then by straightforward computation it follows that ω_{x}(−1) = − exp{−π Im c/ Re c} and ω_{x}(τ) = τ^{1/ Re c} for τ > 0.

Corollary 4. Under the hypothesis of Theorem 2 the property of asymptotical symmetry (2.2) holds if and only if Im c = 0.

Corollary 5. Under the hypothesis of Theorem 1 we have c = 1 and therefore

(2.20)  lim_{{t→0 \atop t∈R\\{0\}}} \frac{f(x + τt) − f(x)}{f(x + t) − f(x)} = τ,  for all τ > 0

uniformly with respect to x ∈ ℳ ⊆ R.

Corollary 6. Let f be a Q-quasiconformal self-mapping of the upper halfplane H with complex dilatation μ such that for fixed x ∈ ℳ and almost all θ, 0 ≤ θ ≤ π,

(2.21)  lim_{t→0} μ(x + te^{iθ}) = ν_{x}(e^{iθ})

uniformly with respect to x ∈ ℳ . Then (2.17)–(2.20) hold also uniformly in ℳ .

It is easy to show that the uniform assumption (2.21) implies the uniform condition (2.16).

We complete this section with a result on boundary correspondence under quasiconformal mappings of the unit disk onto itself. In analogy with the halfplane model, an orientation-preserving self-homeomorphism f of the unit circle S^{1} is called quasisymmetric if it can be extended to a quasiconformal mapping of the unit disk D.

Theorem 3. Let f be a quasiconformal self-mapping of the unit disk D with complex dilatation κ. If there exists a complex number σ_{η} such that

(2.22)  lim_{t→0} \frac{1}{t^{2}} ∫∫_{|ζ−η|≤t,ζ∈D} |κ(ζ) − σ_{η}|^{α} dm_{ζ} = 0

for some α > 0 and fixed η = e^{iφ} ∈ S^{1} , then the quasisymmetric homeomorphism f(e^{iθ}) = e^{iΩ(θ)} satisfies the following condition

(2.23)  lim_{τ→0} \frac{Ω(φ + τ) − Ω(φ)}{Ω(φ) − Ω(φ − τ)} = 1.

Moreover, if the limit (2.22) is uniform with respect to a parameter η ∈ ℳ ⊆ S^{1} then (2.23) is also uniform in ℳ .

Proof. Let f be a quasiconformal self-mapping of the unit disk D. Then

(2.24)  h(z) = −i ln f(e^{iz})

is the corresponding quasiconformal self-mapping of the upper half-plane H for which all hypothesis of Theorem 1 are fulfilled. Indeed, writing f = 𝒜 ∘ h ∘ 𝒜^{−1} , where 𝒜 (z) = e^{iz}, we deduce that

κ(ζ) = (μ\frac{𝒜_{z}}{\overline{𝒜_{z}}} ) ∘ 𝒜^{−1}(ζ).

Thus

μ(z) = −\frac{\overline{e^{iz}}}{e^{iz}} κ(e^{iz})

and therefore

\frac{1}{t^{2}} ∫∫_{|z−x|≤t,z∈H} |μ(z) − ν_{x}|^{α} dm_{z} ∼ \frac{1}{t^{2}} ∫∫_{|ζ−η|≤t,ζ∈D} |κ(ζ) − σ_{η}|^{α}dm_{ζ}

as t → 0. Now assertion (2.23) follows immediately from Theorem 1 being applied to the function h(z).

3. Proof of the lemma

The space 𝔉_{Q} of normalized quasiconformal mappings of the complex plane C with the topology of the locally uniform convergence is metrizable and sequentially compact (see, for instance, [20, p. 71]).

Any metric ϱ defined on 𝔉_{Q} is called a generating metric if the convergence ϱ(f_{n}, f) → 0 is equivalent to the locally uniform convergence f_{n} → f. One of such metrics is (see, e.g. [18])

(3.1)  ϱ(f, g) = ∑_{m=1}^{∞} 2^{−m} \frac{ϱ_{m}(f, g)}{1 + ϱ_{m}(f, g)}

where

(3.2)  ϱ_{m}(f, g) = max_{|z|≤m} |f(z) − g(z)|.

Let 𝔐_{Q} be the space of all complex dilatations μ for mappings f ∈ 𝔉_{Q}. Any metric r defined on 𝔐_{Q} is called majorizing if the convergence r(μ_{n}, ν_{n}) → 0 implies ϱ(f_{n}, g_{n}) → 0 for any generating metric ϱ on 𝔉_{Q} and for the corresponding mappings f_{n} and g_{n} ∈ 𝔉_{Q}.

Proposition 2. Let ϱ and r be generating and majorizing metrics on 𝔉_{Q} and 𝔐_{Q}, respectively. Then r(μ_{t,j}, μ_{j}) → 0 as t → 0 uniformly with respect to an abstract parameter j ∈ J implies the uniform convergence ϱ(f_{t,j}, f_{j}) → 0 in j ∈ J.

Indeed, suppose that there exist an ε > 0 and sequences t_{n} → 0, j_{n} ∈ J, such that ϱ(g_{n}, h_{n}) ≥ ε, where g_{n} = f_{t_{n},j_{n}}, h_{n} = f_{j_{n}}, n = 1, 2, …. However r(μ_{n}, ν_{n}) → 0 and therefore ϱ(g_{n}, h_{n}) → 0. The last statement contradicts the above assumption.

Proposition 3. For some α > 0 let

(3.3)  r(μ, ν) = ∑_{m=1}^{∞} 2^{−m} \frac{r_{m}(μ, ν)}{1 + r_{m}(μ, ν)}

where

(3.4)  r_{m}(μ, ν) = ∫∫_{|z|≤m} |μ(z) − ν(z)|^{α} dm_{z}.

Then r(μ, ν) is a majorizing metric on 𝔐_{Q}.

Proof. Assume that r(μ_{n}, ν_{n}) → 0 but ϱ(f_{n}, g_{n}) ≥ ε > 0 for some generating metric ϱ and some sequences f_{n} and g_{n} ∈ 𝔉_{Q}. We may suppose also that f_{n} → f ∈ 𝔉_{Q}, g_{n} → g ∈ 𝔉_{Q}.

Next we show that f ≡ g. By the Lebesgue convergence theorem r(μ_{n}, ν_{n}) → 0 is equivalent to μ_{n} − ν_{n} ⟶^{mes} 0 on 𝔐_{Q}. Hence

ϕ_{n}(z) = \frac{μ_{n}(z) − ν_{n}(z)}{1 − μ_{n}(z)\overline{ν_{n}(z)}} \frac{(g_{n})_{z}}{\overline{(g_{n})_{z}}} ⟶^{mes} 0.

To continue the proof we need the following principal property of normalized Qquasiconformal mappings related with the area distortion. Let f ∈ 𝔉_{Q} then for each measurable set E ⊂ D_{R}

mes(f(E)) ≤ c(Q, R)(mes(E))^{δ(Q)}

where D_{R} = {z : |z| < R} and the constants c and δ depend only on Q and R and Q, respectively (see, e.g. [5], [11], [14], [19]).

The above result implies that

κ_{n} = ϕ_{n} ∘ g_{n}^{−1} ⟶^{mes} 0

where the κ_{n} represent the complex dilatations of the mappings h_{n} = f_{n} ∘ g_{n}^{−1} ∈ 𝔉_{Q^{2}} . Thus h_{n} → h = f ∘ g^{−1} and simultaneously κ_{n} ⟶^{mes} 0. By the Bers–Bojarski convergence theorem (see [20, p. 187]) we deduce that g = f. Therefore

ϱ(f_{n}, g_{n}) ≤ ϱ(f_{n}, f) + ϱ(g_{n}, f) → 0

as n → ∞. The last contradicts the above assumption.

On boundary correspondence 177

Proof of the lemma. Let f_{j}: C → C, f_{j}(0) = 0, j ∈ J, be a family of Q-quasiconformal mappings with complex dilatations μ_{j}. It is easy to verify that μ_{t,j}(z) = μ_{j}(tz) represent the complex dilatations for the quasiconformal mappings g_{t,j}(z) = f_{j}(tz)/f_{j}(t) ∈ 𝔉_{Q}, t ∈ R \ {0}. Fix any α > 0 and set t = τm, z = τζ, in (2.14). Then we deduce that

lim_{τ→0} ∫∫_{|ζ|≤m} |μ_{τ,j}(ζ) − ν_{j}(ζ)|^{α} dm_{ζ} = 0

uniformly in j ∈ J for all m = 1, 2, …. Now by Propositions 2 and 3 g_{t,j}(z) → g_{j}(z) ∈ 𝔉_{Q} locally uniformly with respect to z ∈ C uniformly in j ∈ J. Here g_{j}(z) has the complex dilatation which agrees with ν_{j}(ζ) almost everywhere. By the uniqueness theorem for the Beltrami equation g_{j}(z) = ω_{j}(z). The lemma is proved.

Acknowledgments. We wish to thank Professor F.W. Gehring and Professor O. Lehto for useful discussions and the referee for useful comments.

The research described in this publication was made possible in part by Grant Number U96000 from the International Science Foundation (ISF).

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Received 4 October 1994