A REMARK ON THE AHLFORS-LEHTO UNIVALENCE CRITERION

TOSHIYUKI SUGAWA

Abstract. In this note, we will prove the Ahlfors-Lehto univalence criterion in a general form. This enables us to deduce lower estimates of the inner radius of univalence for an arbitrary quasidisk in terms of a given quasiconformal reflection.

1. Introduction

Let D be a domain in the Riemann sphere ℂ̂ with hyperbolic metric ρ_{D}(z)|dz| of constant negative curvature −4. For a holomorphic function φ on D, we define the hyperbolic sup-norm of φ by

∥φ∥_{D} = sup_{z∈D} ρ_{D}(z)^{−2}|φ(z)|.

We denote by B_{2}(D) the complex Banach space consisting of all holomorphic functions of finite hyperbolic sup-norm. For a holomorphic map g : D_{1} → D_{2}, the pullback g_{2}^{∗} : φ ↦ φ ∘ g · (g′)^{2} is a linear contraction from B_{2}(D_{2}) to B_{2}(D_{1}). In particular, if g is biholomorphic, g_{2}^{∗} : B_{2}(D_{2}) → B_{2}(D_{1}) becomes an isometric isomorphism. As is well known, the Schwarzian derivative S_{f} = (f′′/f′)′−(f′′/f′)^{2}/2 of a univalent function f on D satisfies ∥S_{f}∥_{D} ≤ 12 (see [3]). This result is classical for the unit disk 𝔻 = {z ∈ ℂ; |z| < 1}, actually, the better estimate ∥S_{f}∥_{𝔻} ≤ 6 is known. On the other hand, Nehari’s theorem [13] asserts that if a locally univalent function f on 𝔻 satisfies ∥S_{f}∥_{𝔻} ≤ 2, then f is necessarily univalent. Hille’s example [7] shows that the number 2 is best possible. We now define the quantity σ(D), which is called the inner radius of univalence of D, as the infimum of the norm ∥S_{f}∥_{D} of those locally univalent meromorphic functions f on D which are not globally univalent in D. In other words, σ(D) is the possible largest number σ ≥ 0 with the property that the condition ∥S_{f}∥_{D} ≤ σ implies univalence of f in D. Note that the inner radius of univalence is Möbius invariant, namely, σ(L(D)) = σ(D) for a Möbius transformation L. In the case D = 𝔻, we already know σ(𝔻) = 2. For a comprehensive exposition of these notions and some background, we refer the reader to the book [9] of O. Lehto.

Date: October 30, 2000.

1991 Mathematics Subject Classification. Primary 30C62; Secondary 30C55, 30F60.

Key words and phrases. Schwarzian derivative, quasiconformal extension, inner radius of univalence.

Ahlfors [1] showed that every quasidisk has positive inner radius of univalence. Conversely, Gehring [6] proved that if a simply connected domain has positive inner radius of univalence then it must be a quasidisk. Later, Lehto [8] pointed out the inner radius of univalence of a quasidisk can be estimated by using Ahlfors’ method as

σ(D) ≥ 2 inf_{z∈D′} \frac{|¯∂λ(z)| − |∂λ(z)|}{|λ(z) − z|^{2}ρ_{D}(z)^{2}} , (1)

where λ is a quasiconformal reflection in ∂D which is continuously differentiable off ∂D and D′ = D\{∞, λ(∞)}. This result may be called the Ahlfors-Lehto univalence criterion. However, in order to obtain estimate (1) rigorously, a kind of approximation procedure must work, so an additional assumption was needed. For example, Lehto [9, Lemma III.5.1] assumed the quasidisk D to be exhausted by domains of the form {rz; z ∈ D} for 0 < r < 1. More recently, Betker [5] gave a similar result for general quasidisks under the assumption that the quasiconformal reflections λ are of a special form associated with the Löwner chains. For another additional condition, see a remark at the end of the next section.

We remark that if we content ourselves with an estimate of the form σ(D) ≥ c(K) for a K-quasidisk D, where c(K) is a positive constant depending only on K, the original idea of Ahlfors [1] is sufficient. (See Section 2. See also [2, Chapter VI] and [9, Theorem II.4.1] for slightly different approaches.)

Our main result is to show (1) without any additional assumption, even the continuous differentiability of λ. This might be known as a kind of folklore. Theorem 1. Let D be a quasidisk with a quasiconformal reflection λ in ∂D. Then the inequality σ(D) ≥ ε(λ, D) holds for D, where

ε(λ, D) = 2 ess. inf_{z∈D} \frac{|¯∂λ(z)| − |∂λ(z)|}{|λ(z) − z|^{2}ρ_{D}(z)^{2}} . (2)

Actually, this estimate is known to give often sharp results for several concrete examples (see [9]). The author, however, does not know if the equality σ(D) = sup_{λ} ε(λ, D) always holds or not, where the supremum is taken over all possible quasiconformal reflections λ in ∂D.

Our main theorem has applications to lower estimates of the inner radius of univalence for a strongly starlike domain (see [15]) and for a round annulus (see [14]). Indeed, the quasiconformal reflections used in those papers are not necessarily of class C^{1} off the boundary.

Finally, the author would like to express his sincere thanks to Professor F. W. Gehring, whose suggestion improved the statement of our Main Theorem.

2. Proof of main result

Basically, we shall go along the same line as in [1]. Let a quasiconformal reflection λ in ∂D be given, i.e., λ is an orientation-reversing homeomorphic involution of ℂ̂ keeping each boundary point of D fixed and satisfying that λ(z̄) is quasiconformal. We note that |∂λ| ≤ k_{0}|¯∂λ| a.e. for some constant 0 ≤ k_{0} < 1.

Noting that the quantity ε(λ, D) is invariant under the Möbius transformations (see [9, Sec. II 4.1]), we assume that a quasidisk D is contained in ℂ for a moment. We take a non-constant meromorphic function f on D with ∥S_{f}∥_{D} < ε_{0} = ε(λ, D). We wish to show that f is univalent in D. Set φ = S_{f}.

Let η_{0} and η_{1} be linearly independent solutions of the linear differential equation 2y′′ + φy = 0 (3)

in D. Note that the Wronskian η_{0}η′_{1} − η′_{0}η_{1} is a non-zero constant and, as is well known, η_{1}/η_{0} satisfies the Schwarzian differential equation S_{η_{1}/η_{0}} = φ = S_{f} in D. In particular, there exists a Möbius transformation L satisfying η_{1}/η_{0} = L ∘ f. Therefore, when we try to show the univalence of f, we can assume, and always do so in the sequel, f = η_{1}/η_{0} and η_{0}η′_{1} − η′_{0}η_{1} ≡ 1.

For instance, if η_{0} and η_{1} are taken by the solutions of (3) satisfying the initial conditions η_{0} = 1, η′_{0} = 0 and η_{1} = 0, η′_{1} = 1, respectively, at a reference point z_{0} in D, then f = η_{1}/η_{0} is strongly normalized at z_{0}: f(z_{0}) = f′(z_{0}) − 1 = f′′(z_{0}) = 0.

To extend f to the whole sphere, we consider the map

F (z) = \frac{η_{1}(z) + (λ(z) − z)η′_{1}(z)}{η_{0}(z) + (λ(z) − z)η′_{0}(z)}.

We first note the Möbius invariance of the above construction. For an A = (_{c}^{a}_{d}^{b}) ∈ SL(2, ℂ) let L_{A} be the Möbius transformation induced by the matrix A. We set A_{2}^{∗}φ = (φ∘ L_{A})(L′_{A})^{2} and A_{−1/2}^{∗}η = (η ∘ L_{A})(L′_{A})^{−1/2}, where (L′_{A})^{−1/2}(z) = cz + d. A straightforward calculation shows that A_{−1/2}^{∗}η is a solution of the differential equation 2y′′ + A_{2}^{∗}φy′′ = 0 in A^{−1}(D) if η is a solution of (3) in D. In particular, we can see that differential equation (3) always admits two linearly independent (single-valued) solutions in D even if ∞ ∈ D. Setting λ̃ = L_{A}^{−1} ∘ λ ∘ L_{A}, we consider the map

F̃(z) = \frac{A_{−1/2}^{∗}η_{1}(z) + (λ̃(z) − z)(A_{−1/2}^{∗}η_{1})′(z)}{A_{−1/2}^{∗}η_{0}(z) + (λ̃(z) − z)(A_{−1/2}^{∗}η_{0})′(z)}.

Then we have the relation F ∘ L_{A} = F̃. In fact, using the relation L_{A}(w) − L_{A}(z) = (w − z)/(cw + d)(cz + d), we obtain

η ∘ L_{A}(z) + (λ ∘ L_{A}(z) − L_{A}(z))η′ ∘ L_{A}(z)

=\frac{A_{−1/2}^{∗}η(z)}{cz + d} + (L_{A}(λ̃(z)) − L_{A}(z))(cz + d)((A_{−1/2}^{∗}η)′(z) − cη ∘ L_{A}(z))
=\frac{A_{−1/2}^{∗}η(z)}{cz + d} + \frac{λ̃(z) − z}{cλ̃(z) + d} ((A_{−1/2}^{∗}η)′(z) − \frac{cA_{−1/2}^{∗}η(z)}{cz + d})
=\frac{A_{−1/2}^{∗}η(z) + (λ̃(z) − z)(A_{−1/2}^{∗}η)′(z)}{cλ̃(z) + d}.

Taking η_{1} and η_{0} as the above η, we see the desired relation.

Next, we need the following fundamental property of F.

Lemma 2. The map F : D → ℂ̂ constructed above is K-anti-quasiregular, where K = (1 + k)/(1 − k), k = 1 − (1 − k_{0})(1 − k_{1}) < 1, and k_{1} = ∥φ∥_{D}/ε_{0} < 1.

Proof. By the Möbius invariance of the construction of F, we may assume here that ∞ ∈ ∂D. We note that the numerator and the denominator in the definition of F can never vanish simultaneously because of the relation η_{0}η′_{1}−η′_{0}η_{1} ≡ 1. Since K-anti-quasiregularity is a local property, it is enough to show that F is K-anti-quasiregular in a neighbourhood of an arbitrary point, say a, in D. We may assume that η_{0}(a) + (λ(a) − a)η′_{0}(a) ≠ 0. (If not, consider 1/F instead.) By continuity, we can take an open neighbourhood V of a in D such that η_{0} + (λ − z)η′_{0} does not vanish at any point of V.

Here, we recall that a non-constant continuous function h : V → ℂ is K-anti-quasiregular if and only if h is ACL and has locally square integrable partial derivatives satisfying |∂h| ≤ k|¯∂h| a.e. in V (see [10, Chapter VI], where the authors used the term “quasiconformal function” instead of “quasiregular mapping”).

Now we show that F is ACL in V, precisely, for any closed rectangle {x + iy; x_{0} ≤ x ≤ x_{1}, y_{0} ≤ y ≤ y_{1}} contained in V, F (x + iy) is absolutely continuous in x ∈ [x_{0}, x_{1}] for a.e. y ∈ [y_{0}, y_{1}] and in y ∈ [y_{0}, y_{1}] for a.e. x ∈ [x_{0}, x_{1}]. Since η_{j} + (λ − z)η′_{j} is absolutely continuous in x ∈ [x_{0}, x_{1}] for a.e. y and for j = 0, 1, and since η_{0} + (λ − z)η′_{0} does not vanish there, we can conclude that the quotient F (x + iy) is also absolutely continuous in x ∈ [x_{0}, x_{1}] for a.e. y ∈ [y_{0}, y_{1}] and in y ∈ [y_{0}, y_{1}] for a.e. x ∈ [x_{0}, x_{1}] (see, for example, [12, p.50]). Hence, F is ACL in V.

Next, we investigate the partial derivatives of F. A formal calculation gives us

∂F = \frac{∂λ + (λ − z)^{2}φ/2}{(η_{0} + (λ − z)η′_{0})^{2}} and ¯∂F = \frac{¯∂λ}{(η_{0} + (λ − z)η′_{0})^{2}} .

Since λ has locally square integrable partial derivatives in D and since the denominator is locally bounded away from 0 in V, we can observe that ∂F and ¯∂F are both locally square integrable in V. Furthermore, we have

\frac{∂F (z)}{¯∂F (z)} = \frac{∂λ(z) + (λ(z) − z)^{2}φ(z)/2}{¯∂λ(z)}.

Hence the assumption ∥φ∥_{D} = ε_{0}k_{1} implies

∥∂F/¯∂F ∥_{∞} ≤ 1 − (1 − k_{0})(1 − k_{1}) = k.

Hence, we have shown that F is K-anti-quasiregular in V.

Now we consider the map f̂: D ∪ D^{∗} → ℂ̂ defined by

f̂(z) = \begin{cases}f(z) & for z ∈ D, \\ F (λ(z)) & for z ∈ D^{∗}, \end{cases} (4)

where D^{∗} = ℂ̂ \ \overline{D}. It is not so clear that f̂ can be extended to ∂D continuously because φ can not be extended to ∂D or beyond it in general. In order to overcome this difficulty, we approximate φ by better quadratic differentials. In fact, for a general φ ∈ B_{2}(D), we have the following result, which is essentially due to Bers [4, Lemma 1].

Proposition 3. Let D be a Jordan domain in ℂ̂. For any φ ∈ B_{2}(D) there exists a sequence (φ_{j})_{j} of holomorphic functions in \overline{D} such that ∥φ_{j}∥_{D} ≤ ∥φ∥_{D} and φ_{j} tends to φ uniformly on each compact subset of D as j → ∞.

Proof. We denote by g : 𝔻 → D the Riemann mapping function of D with g(0) = z_{0} and g′(0) > 0. Let D_{j}, j = 1, 2, … be Jordan domains with \overline{D}_{j+1} ⊂ D_{j} and with ⋂_{j} D_{j} = D. Then the Carathéodory kernel theorem implies that the Riemann mapping functions g_{j} of D_{j} with g_{j}(0) = z_{0} and g′_{j}(0) > 0 converge to g uniformly on each compact subset of the unit disk as j tends to ∞. Now we set φ_{j} = (g ∘ g_{j}^{−1})_{2}^{∗}φ. We then have ∥φ_{j}∥_{D} ≤ ∥φ_{j}∥_{D_{j}} = ∥φ∥_{D} by the Schwarz-Pick lemma: ρ_{D} ≥ ρ_{D_{j}}. We also have φ_{j} → φ locally uniformly as j → ∞.

With this result in mind, we can deduce our main result from the following lemma. Lemma 4. Suppose that φ ∈ B_{2}(D) with ∥φ∥_{D} ≤ k_{1}ε_{0} is holomorphic in \overline{D}, where 0 ≤ k_{1} < 1 and ε_{0} = ε(λ, D), which is given by (2). Then the function f̂ defined by (4) extends to a K-quasiconformal homeomorphism of the Riemann sphere, where K = (1 + k)/(1 − k) and k = 1 − (1 − k_{0})(1 − k_{1}).

Actually, we can prove our main theorem as follows. Let φ ∈ B_{2}(D) satisfy ∥φ∥_{D} < ε_{0} and set k_{1} = ∥φ∥_{D}/ε_{0}. We take a sequence (φ_{j})_{j} as in Proposition 3. Let f̂ and f̂_{j} be the functions in ℂ̂ \ ∂D defined by (4) for φ and φ_{j}, respectively, so that both are strongly normalized at z_{0} ∈ D. Then, by the above lemma, each f̂_{j} can be continued to a K-quasiconformal homeomorphism of ℂ̂. Since those K-quasiconformal mappings which are conformal in D and strongly normalized at z_{0} form a normal family, f̂_{j} has a subsequence converging to a K-quasiconformal mapping uniformly in ℂ̂. By construction, the limit mapping coincides with f̂ in ℂ̂\∂D. This implies that f̂ has a K-quasiconformal extension to the whole sphere. Now the proof of our main theorem is complete up to the above lemma.

Remark. Under the assumption that λ is of class C^{1} off the boundary ∂D and that φ is holomorphic in \overline{D} with ∥φ∥_{D} < ε_{0}, a direct calculation shows

∂f̂(z) = −\frac{1 + (z − λ(z))^{2}φ(λ(z))∂λ(z)/2}{(η_{0}(λ(z)) + (z − λ(z))η′_{0}(λ(z)))^{2}} and
¯∂f̂(z) = −\frac{(z − λ(z))^{2}φ(λ(z))¯∂λ(z)/2}{(η_{0}(λ(z)) + (z − λ(z))η′_{0}(λ(z)))^{2}}

at every z ∈ D^{∗} \ {∞, λ(∞)}. Therefore, if (λ(z) − z)^{2} ¯∂λ(z) vanishes at the boundary, then we would obtain continuous extensions of ∂f̂ and ¯∂f̂ to ℂ̂. Moreover, the limits of

\frac{f̂(z + t) − f̂(z)}{t} and \frac{f̂(z + it) − f̂(z)}{t}

when t tending to 0 along the real axis both exist and are equal to f′(z) and if′(z), respectively, for each z ∈ ∂D. In fact, when z + t or z + it approaches to z in D^{∗}, the above quotients tend to the desired values by (7) below. This implies that our f̂ has continuous partial derivatives everywhere in ℂ̂. Hence, we can conclude that f̂ is a local C^{1}-diffeomorphism of ℂ̂, and hence, a global C^{1}-diffeomorphism of it. Thus, if we restrict ourselves to this case, the proof would become much simpler than ours.

We note that it is always possible to take such a quasiconformal reflection λ for any quasidisk D (see [1] or [9, Section II.4]).

3. Proof of Lemma 4

Let φ be as in Lemma 4. We assume that D is bounded for a moment. Then the solutions η_{0} and η_{1} of (3) are holomorphic in \overline{D}. Thus f̂ can be continuously extended to the whole sphere and f̂(∂D) is the image of the quasicircle ∂D under the locally univalent meromorphic map η_{1}/η_{0}. Now we require an extension theorem for quasiregular mappings.

Lemma 5. Let Ω be a plane domain and C be an open quasiarc (or a quasicircle) in Ω such that Ω \ C is an open set in ℂ̂. Suppose that h : Ω → ℂ̂ is a continuous map such that h|_{Ω\C} is a K-quasiregular map and that, for each x ∈ C, h maps C ∩ U injectively onto a quasiarc for some open neighbourhood U of x in Ω. Then h is K-quasiregular in Ω.

Proof. If once we know that h is quasiregular in Ω, we can conclude that h is Kquasiregular because |¯∂h/∂h| ≤ (K −1)/(K +1) a.e. by assumption. Since quasiregularity is a local property, it suffices to show that h is quasiregular in an open neighbourhood U of each x ∈ C. The assumption allows us to take an open neighbourhood U of x so that h maps U ∩ C injectively onto a quasiarc. Then, by composing suitable quasiconformal mappings, we may further assume that U is an open disk centered at x = 0 with U ∩ C = U ∩ ℝ and that h(U ∩ ℝ) ⊂ ℝ. Set U_{±} = {z ∈ U; ±Im z ≥ 0}. By the reflection principle for quasiregular mappings [11], the mapping h|_{U_{±}} extends to a quasiregular one in U for each signature. This means that h is ACL and has locally square integrable partial derivatives in U, and hence h is quasiregular there.

By this lemma, our mapping f̂ turns out to be a K-quasiregular mapping on ℂ̂, and hence, f̂ can be decomposed to the form g ∘ω for a K-quasiconformal mapping ω : ℂ̂ → ℂ̂ and a rational function g (see [10, Chapter VI]). Suppose that the degree of g is greater than one. Then there exists a branch point, say b^{∗}, of g. Set a^{∗} = ω^{−1}(b^{∗}) and a = λ(a^{∗}). In this time, by Möbius conjugation, we assume that 0, ∞ ∈ ∂D and the branch points of f̂ and their reflections under λ are all finite.

Since f̂ is locally injective in D, the point a^{∗} must lie in ℂ̂ \ D, thus a ∈ \overline{D}.

First, we show that a ∉ D. Suppose that a ∈ D. By assumption, note that η_{0}(a) + (λ(a) − a)η′_{0}(a) ≠ 0. We now investigate the local behaviour of F near the point a. Since a^{∗} is a branch point of F ∘ λ, the image of the positively oriented loop |z − a| = r under F would have winding number N with N < −1 around F (a) for a sufficiently small r > 0. Setting δ = z − a and δ^{∗} = λ(z) − λ(a), we have

η_{j}(z) + (λ(z) − z)η′_{j}(z) = η_{j}(a) + η′_{j}(a)δ + (λ(a) − a + δ^{∗} − δ)(η′_{j}(a) + η′′_{j}(a)δ) + O(δ^{2})
= η_{j}(a) + (λ(a) − a + δ^{∗})η′_{j}(a) + (λ(a) − a)η′′_{j}(a)δ + o(δ)

as δ → 0 for j = 0, 1. Using the relations η_{0}η′_{1} − η′_{0}η_{1} = 1 and η′′_{j} = −φη_{j}/2 also, we calculate

F (z) − F (a) = \frac{η_{1}(z) + (λ(z) − z)η′_{1}(z)}{η_{0}(z) + (λ(z) − z)η′_{0}(z)} − \frac{η_{1}(a) + (λ(a) − a)η′_{1}(a)}{η_{0}(a) + (λ(a) − a)η′_{0}(a)}

= \frac{φ(a)(λ(a) − a)^{2}δ/2 + δ^{∗} + o(δ)}{(η_{0}(a) + (λ(a) − a)η′_{0}(a))^{2} + o(1)}

= \frac{λ(z) − λ(a) + c(z − a) + o(z − a)}{(η_{0}(a) + (λ(a) − a)η′_{0}(a))^{2} + o(1)}

as z → a, where c = φ(a)(λ(a) − a)^{2}/2. From ∥φ∥_{D} ≤ k_{1}ε_{0}, we deduce

|c| ≤ k_{1} (\frac{|λ(a) − a|ρ_{D}(a)}{|λ(z) − z|ρ_{D}(z)})^{2} (|¯∂λ(z)| − |∂λ(z)|)

for almost all z ∈ D. So, if we are given a number s with k_{1} < s < 1, then we can find a sufficiently small number r so that

|c| < s · ess. inf_{𝔻(a,r)}(|¯∂λ| − |∂λ|) (5)

holds, where 𝔻(a, r) = {z ∈ ℂ; |z − a| < r}. We now need the following fact about the local behaviour of quasiconformal maps, which might be interesting in itself.

Lemma 6. Let h : ℂ → ℂ be a quasiconformal homeomorphism. For a point a ∈ ℂ and a radius r > 0, suppose that E = ess. inf_{𝔻(a,r)}(|∂h| − |¯∂h|) > 0. Then, the map h_{t}(z) = h(z) + tz̄ is quasiconformal on the disk 𝔻(a, (1 − s)r) for any t ∈ ℂ with |t| < sE. Furthermore, we have

|h_{t}(z) − h_{t}(a)| ≥ (sE − |t|)|z − a| for z ∈ 𝔻(a, (1 − s)r). (6)

We postpone the proof to Section 4 because we require some trick to show this.

Set H(z) = (λ(z) + cz)/(η_{0}(a) + (λ(a) − a)η′_{0}(a))^{2}. Using (5), we now apply the above lemma to the case h = λ̄ and t = c̄ and see that H is anti-quasiconformal near the point a. In particular, the image of the positively oriented loop ℓ_{r} : |z − a| = r under H has winding number −1 around the point H(a) for r small enough. Now estimate (6) guarantees that F (z) − F (a) = (H(z) − H(a))(1 + o(1)) as z → a, which implies that the winding number of the image of ℓ_{r} under F around the point F (a) = H(a) is equal to that of H for sufficiently small r. This contradicts the fact that a is a branch point of F. We now conclude that a ∉ D.

Therefore, the point a^{∗} must lie in ∂D if f̂ has a branch point a^{∗}. Since a = λ(a^{∗}) = a^{∗} in this case, we may use the letter a instead of a^{∗}.

We will use the following important fact on quasiconformal reflections to deduce a contradiction.

Lemma 7 ([9, Lemma I.6.3]). Let λ be a K-quasiconformal reflection in C with ∞ ∈ C. Then

\frac{1}{M(K)}|z − ζ| ≤ |λ(z) − ζ| ≤ M(K)|z − ζ|

for any z ∈ ℂ and ζ ∈ C, where M(K) > 1 is a constant depending only on K.

The map f̂ is never injective near a while f̂|_{D} = f = η_{1}/η_{0} is injective near a, so we can select sequences of pairs of points z_{n} and w_{n} in D and closed arcs α_{n} connecting z_{n} and w_{n} in D such that F (z_{n}) = F (w_{n}) and F (α_{n}) has winding number ±1 around F (a) = f(a), and that z_{n} → a, w_{n} → a and diam α_{n} → 0 as n → ∞, where diam stands for the Euclidean diameter. Here and hereafter, we always understand that curves are parametrized by the standard interval [0, 1].

Now we consider the asymptotic behaviour of F (z) as z tends to a in D. Keeping Lemma 7 in mind, in the same way as above, we can show that

F (z) − F (a) = \frac{η_{1}(z) + (λ(z) − z)η′_{1}(z)}{η_{0}(z) + (λ(z) − z)η′_{0}(z)} − \frac{η_{1}(a)}{η_{0}(a)}

= η_{0}(a)^{−2}(λ(z) − a) + O((z − a)^{2}) (7)

as z → a in D.

Therefore, combining with Lemma 7, we have F (α_{n}(t)) − F (a) − η_{0}(a)^{−2}(α_{n}^{∗}(t) − a) = O((α_{n}(t) − a)^{2}) = O((α_{n}^{∗}(t) − a)^{2}) uniformly in t as n → ∞, where α_{n}^{∗}(t) = λ(α_{n}(t)). In particular, η_{0}(a)^{2}(F (α_{n}(t)) − F (a))/(α_{n}^{∗}(t) − a) = 1 + o(1), hence |F (α_{n}(t)) − F (a) − η_{0}(a)^{−2}(α_{n}^{∗}(t) − a)| < |F (α_{n}(t)) − F (a)| (8)

holds in t ∈ [0, 1] for sufficiently large n.

Now we may assume |z_{n}| ≥ |w_{n}| for every n. Since F (z_{n}) = F (w_{n}) we have δ_{n} := |z_{n}^{∗} − w_{n}^{∗}| = O(|z_{n} − a|^{2}) as n → ∞ by (7), where we set z_{n}^{∗} = λ(z_{n}) and w_{n}^{∗} = λ(w_{n}).

Here, we recall a fundamental property of quasidisks. The linear connectedness of D^{∗} asserts the existence of a constant M > 1 such that any pair of points in D^{∗} ∩ 𝔻(c, r) can be joined by a curve in D^{∗} ∩ 𝔻(c, Mr) for all c ∈ ℂ and r > 0 (see [6] or [9]). In particular, there exists a sequence of curves β_{n}^{∗} connecting w_{n}^{∗} and z_{n}^{∗} in D^{∗} ∩ 𝔻(z_{n}^{∗}, Mδ_{n}). Therefore we have |(F (z_{n}) − F (a)) − η_{0}(a)^{−2}(β_{n}^{∗}(t) − a)| ≤ M|η_{0}(a)|^{−2}δ_{n} + O(|z_{n} − a|^{2}) = O(|z_{n} − a|^{2}) = O(|z_{n}^{∗} − a|^{2}) = O(|F (z_{n}) − F (a)|^{2}) as n → ∞, and then |(F (z_{n}) − F (a)) − η_{0}(a)^{−2}(β_{n}^{∗}(t) − a)| < |F (z_{n}) − F (a)| (9)

for n large enough.

Now we conclude from (8) and (9) that the closed curves F (α_{n}) − F (a) and γ_{n}^{∗} − a, where γ_{n}^{∗} := α_{n}^{∗} · β_{n}^{∗}, have the same winding number around 0 for sufficiently large n. By the choice of α_{n}, we see that γ_{n}^{∗} has winding number ±1, and hence γ_{n}^{∗} separates a from ∞ for such an n. Since a and ∞ belong to ∂D^{∗} and since γ_{n}^{∗} is a curve in D^{∗}, this situation contradicts the fact that D^{∗} is simply connected. This contradiction is caused by the assumption deg g > 1. Therefore we can now conclude that g is a Möbius transformation, and hence the proof of Lemma 4 is now complete except for Lemma 6.

4. Proof of Lemma 6

Set k_{0} = ∥¯∂h/∂h∥_{∞} < 1. Without loss of generality, we can assume that a = 0. We simply write 𝔻(r) = 𝔻(0, r). First note that h_{t}(z) = h(z) + tz̄ is quasiregular in 𝔻(r) for |t| < E. In fact, |¯∂h_{t}/∂h_{t}| ≤ (|∂λ| + |t|)/|¯∂λ| ≤ k_{0} + k_{1} − k_{0}k_{1} = 1 − (1 − k_{0})(1 − k_{1}) < 1 a.e. in 𝔻(r) if |t|/E = k_{1} < 1.

Put ε = 1 − s. Now we use the auxiliary function χ : ℂ → ℂ which is defined by

χ(z) = \begin{cases}z̄ & if |z| ≤ εr \\ ε(1 − ε)^{−1}(r − |z|)z̄/|z| & if εr ≤ |z| ≤ r \\ 0 & if r ≤ |z|. \end{cases}

Then we can extend h_{t}|_{𝔻(εr)} to the complex plane, which will be still denoted by the same letter, by the relation h_{t} = h + tχ. Since

∂χ(z) = −\frac{εr}{2(1 − ε)} · \frac{|z|}{z^{2}} and ¯∂χ(z) = \frac{εr}{2(1 − ε)} (\frac{1}{|z|} − \frac{2}{r}) ,

we have |∂χ| ≤ 1/2(1 − ε) = 1/2s and |¯∂χ| ≤ max{ε, |1 − 2ε|}/2(1 − ε) < 1/2s on the annulus {εr < |z| < r}. Setting k = |t|/sE < 1, we see that

|\frac{¯∂h_{t}}{∂h_{t}} | ≤ \frac{|¯∂h| + |t|/2s}{|∂h| − |t|/2s} ≤ \frac{2|¯∂h| + k(|∂h| − |¯∂h|)}{2|∂h| − k(|∂h| − |¯∂h|)} ≤ \frac{m + k_{0}}{1 + mk_{0}} < 1

holds a.e. in the above annulus, where m = k/(2 − k) < 1. Combining this with the fact that h_{t} is quasiregular in |z| < εr and in |z| > r, we can see that h_{t} is quasiregular in ℂ for each t ∈ 𝔻(sE). Set μ_{t} = ¯∂h_{t}/∂h_{t} and let ω_{t} be the quasiconformal automorphism of ℂ satisfying the partial differential equation ¯∂ω_{t} = μ_{t}∂ω_{t} a.e. on ℂ and the normalization ω_{t}(0) = 0 and ω_{t}(1) = 1. Then Q_{t} = h_{t} ∘ ω_{t}^{−1} is an entire function for each t ∈ 𝔻(sE). Since h_{t} = h near the point at infinity, Q_{t} can be holomorphically extended to ∞ so that Q_{t}^{−1}(∞) = {∞} and that Q_{t} is locally biholomorphic near ∞. In particular, Q_{t} is a polynomial of degree 1, and hence an analytic automorphism of ℂ. Thus we can conclude that h_{t} = Q_{t}^{−1} ∘ ω_{t} is also a quasiconformal map of ℂ. Since h_{t}(z) = h(z) + tz̄ for z ∈ 𝔻(εr) = 𝔻((1 − s)r), the first assertion in Lemma 6 now follows.

The latter part of Lemma 6 can immediately be deduced from the former one. Indeed, for each fixed z ∈ 𝔻((1−s)r) other than 0 and for t ∈ 𝔻(sE), the fact that h(z)+(t+u)z̄ = h_{t}(z) + uz̄ never vanishes whenever |t| + |u| < sE implies that |h_{t}(z)| ≥ (sE − |t|)|z̄| = (sE − |t|)|z|.

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Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan Current address: Department of Mathematics, University of Helsinki, P.O.Box 4 (Yliopistonkatu 5),

FIN-00014, Helsinki, Finland

 E-mail address: sugawa@kusm.kyoto-u.ac.jp