HOLOMORPHIC MOTIONS AND QUASICONFORMAL EXTENSIONS

TOSHIYUKI SUGAWA

Abstract. In this article, we will consider holomorphic families of univalent functions parametrized by the unit disk in such a way that the origin corresponds to the identity. For each member of the family, thanks to the λ-lemma, we then have a quasiconformal extension of it. This method enables us not only to give simple proofs of some known results, but also to provide new results. Actually, we derive several results about quasiconformal extendability for typical classes of univalent functions.

1. Introduction

Univalence criteria sometimes produce quasiconformal extension criteria. A number of authors obtained such a kind of results by means of Grunsky’s inequality or Löwner’s method (see, for example, [6] for a comprehensive account, and see also [2]). Grunsky’s inequality does not yield an explicit bound for the maximal dilatations of the quasiconformal extensions in general, however Löwner’s method is sometimes satisfactory but difficult to use. On the other hand, the so-called λ-lemma and its variants generate quantitative results on quasiconformal extendability in various situations, which are often sharp. But, we should notice the reader that this method has the disadvantage that the quasiconformal extension cannot be given explicitly in many cases.

In this article, we shall demonstrate the power of the λ-lemma. First, compare the following two results.

Theorem A (Krzyż [15], cf. [21] p.294). Let ω be an analytic function on the unit disk Δ with |ω′(z)| ≤ k, where 0 ≤ k < 1 is a constant. Then the function f(z) = z + ω(1/z) on the outside of Δ can be extended to a k-quasiconformal automorphism of the Riemann sphere by setting f(z) = z + ω(z̄) on Δ.

Theorem B (Fait, Krzyż and Zygmunt [12, Theorem 2’]). Let ω be an analytic function on the unit disk Δ with |ω′(z)| ≤ k, where 0 ≤ k < 1 is a constant. Then the function f(z) = z+ω(z) on Δ can be extended to a k-quasiconformal automorphism of the Riemann sphere by setting f(z) = z + ω(1/z̄) outside Δ.

As the function f(z) = z + k/z shows, Theorem A is best possible, however Theorem B can be improved by using the λ-lemma as follows: Under the same hypothesis as in Theorem B, the function f(z) = z + ω(z) can be extended to a k′-quasiconformal automorphism of the Riemann sphere, where k′ = k/(2 − k) < k. In fact, this can be easily obtained as a corollary of the following result, but a way of k′-quasiconformal extension of f is not clear from our method.

Date: November 18, 1998.

Key words and phrases. holomorphic motion, quasiconformal extension, univalent function .

Theorem 1.1. Let k be a constant in [0, 1). For an analytic function f on the unit disk Δ with f(0) = 0 and f′(0) ≠ 0, let p(z) represent one of the quantities zf′(z)/f(z), 1 + zf′′(z)/f′(z) or f′(z). If |p(z) − (1 + k^{2})/(1 − k^{2})| ≤ 2k/(1 − k^{2}) for all z ∈ Δ, the function f can be extended to a k-quasiconformal automorphism of the Riemann sphere.

In the case that p(z) = zf′(z)/f(z) or f′(z) the above result is best possible as the function p(z) = (1 + kz^{2})/(1 − kz^{2}) indicates (see Section 4 for details).

The remainder of this paper will be organized as follows. In Section 2, we present a precise definition of holomorphic motions and state the λ-lemma and its consequences. We also introduce (the Bers embedding of) the universal Teichmüller space and recall its fundamental properties. In Section 3, we shall explain a general principle strengthening univalence criteria into quasiconformal extension ones thanks to the λ-lemma. Section 4 is devoted to applications of the general principle to several concrete cases. Consequently, we have a number of quasiconformal extension criteria, most of which seem not to appear in the literature. From these, Theorem 1.1 above immediately follows.

2. Holomorphic motions and the universal Teichmüller space

We recall here the definition of holomorphic motions and its properties. Let Δ denote the unit disk in the complex plane. A holomorphic motion of the subset E of the Riemann sphere ℂ̂ is a map F : E × Δ → ℂ̂ satisfying the following three condtions.

1. For each z ∈ E, the map F (z, ·) : Δ → ℂ̂ is holomorphic,

2. For each λ ∈ Δ, the map F_{λ} := F (·, λ) : E → ℂ̂ is injective, and

3. F_{0} = id_{E}.

The notion of holomorphic motions was first introduced by Mañé, Sad and Sullivan [18] in order to investigate the complex dynamics on the Riemann sphere, and nowadays proves to be very useful in various aspects. The following results will be fundamental in our argument.

Theorem C (Mañé-Sad-Sullivan [18] and Bers-Royden [7]). Let F : E × Δ → ℂ̂ be a holomorphic motion of E. Then the following hold.

1. The map F is uniformly jointly continuous in two variables. Therefore, F uniquely extends to a holomorphic motion of \overline{E}, which will be denoted still by the same letter.

2. For each λ ∈ Δ, the map F_{λ} is quasiconformal in the interior Int(\overline{E}) of \overline{E}.

3. The Beltrami coefficient μ(λ) = ¯∂F_{λ}/∂F_{λ} is a holomorphic map from Δ to the unit ball of the complex Banach space L^{∞}(Int(\overline{E})).

The next striking result was first established by Słodkowski [22]. For another proof, see also Astala-Martin [4] and Douady [9].

Theorem D. Every holomorphic motion of E can be extended to a holomorphic motion of the whole sphere ℂ̂.

We should note that the above extension is not necessarily unique. These two theorems have various applications to the Teichmüller theory (see [11]).

Now we remind the reader of the definition of the universal Teichmüller space and relavant notions (see, as a general reference, [17]). Let B_{j}(D) (j = 1, 2) denote the complex Banach space consisting of all holomorphic functions φ on the hyperbolic domain D in ℂ̂ with hyperbolic (Poincaré) metric ρ_{D}(z)|dz| of constant negative curvature −4 such that the norms

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

are finite. The universal Teichmüller space T (Δ) is the set of Schwarzian derivatives S_{f} = (f′′/f′)′ − (f′′/f′)^{2}/2 of those univalent functions f on the unit disk which can be quasiconformally extended to the whole sphere. The famous Nehari-Kraus theorem claims that ∥S_{f}∥_{2,Δ} ≤ 6 for a univalent function f on Δ, hence T (Δ) ⊂ B_{2}(Δ). Moreover, by Ahlfors, T (Δ) is shown to be a bounded contractible domain of B_{2}(Δ).

We also note that ∥T_{f}∥_{1,Δ} ≤ 6 holds for a univalent function f, where T_{f} denotes the pre-Schwarzian derivative f′′/f′ of f. Thus, the set T_{1}(Δ) of T_{f} of all univalent functions f : Δ → ℂ which admit quasiconformal extensions to the whole sphere is sometimes thought to be another model of the universal Teichmüller space (cf. [3] and [24]).

The Teichmüller distance between pair of points S_{f} and S_{g} of T (Δ) is defined by

d_{T} (S_{f}, S_{g}) = inf_{h} d_{Δ}(0, ∥μ_{h}∥_{∞}),

where h runs over all quasiconformal automorphisms of ℂ̂ such that h = f ∘ g^{−1} on g(Δ) with Beltrami coefficient μ_{h} = hz̄/h_{z} and d_{Δ} denotes the hyperbolic distance determined by the hyperbolic metric ρ_{Δ}(z)|dz| = |dz|/(1 − |z|^{2}) on Δ, i.e., d_{Δ}(z, w) = arctanh(|z − w|/|1 − z̄w|). We note that the infimum is always attained by some h. In particular, we note that the quantity d_{T} (S_{f}, 0) measures the smallest maximal dilatation of the quasiconformal extension of f. A quasiconformal extension f̃ to ℂ̂ of f is called extremal if the Beltrami coefficient μ of f̃ satisfies d_{T} (0, S_{f}) = d_{Δ}(0, ∥μ∥_{∞}). It is known that f̃ is extremal if and only if μ satisfies the Hamilton-Krushkal’ condition (cf. Gardiner [13] Chapter 6):

sup_{φ} |∫∫ _{Δ^{∗}} μ(z)φ(z)dxdy| = ∥μ∥_{∞},

where the supremum is taken over all integrable holomorphic quadratic differentials φ = φ(z)dz^{2} on the exterior Δ^{∗} of the unit disk with ∥φ∥ = ∫∫_{Δ^{∗}} |φ(z)|dxdy ≤ 1. Note that the Hamilton-Krushkal’ condition is conformally invariant.

The cerebrated Royden-Gardiner theorem states that the Teichmüller distance d_{T} coincides with the Kobayashi (pseudo-)distance on T (Δ) (for a modern proof also using the optimal λ-lemma, see [11]). By the contraction property of Kobayashi pseudo-distance, we have the following (known) result as a simple corollary.

Proposition 2.1. Suppose that h : Δ → T (Δ) is a holomorphic map. Then we have d_{T} (h(s), h(t)) ≤ d_{Δ}(s, t) for any s, t ∈ Δ.

Let Φ : M_{1} → T (Δ) denote the Bers projection, where M_{k} = {μ ∈ L^{∞}(ℂ); ∥μ∥_{∞} < k, μ|_{Δ} = 0}. Precisely speaking, Φ(μ) is the Schwarzian derivative of w^{μ}|_{Δ}, where w^{μ} is a homeomorphic solution of the Beltrami equation wz̄ = μw_{z} on ℂ̂ (note that w^{μ} is conformal on Δ by the assumption μ = 0 on Δ). As is well-known, the map Φ : M_{1} → T (Δ) is a holomorphic submersion.

The λ-lemma has an important application to the Teichmüller theory. Now we explain it. Let h : Δ → T (Δ) be a holomorphic map. We choose a μ_{0} ∈ M_{1} such that Φ(μ_{0}) = h(0) and fix it. For each λ ∈ Δ, a meromorphic function f_{λ} on Δ is uniquely determined by the conditions: (1) S_{f_{λ}} = h(λ), (2) f_{λ}(0) = f′_{λ}(0) − 1 = f′′_{λ}(0) = 0. In such a way, we obtain a holomorphic motion F (z, λ) = f_{λ} ∘ f_{0}^{−1}(z) of f_{0}(Δ). By Theorem D, F can be extended to a holomorphic motion F̃ of the whole sphere. By assumption, the univalent function f_{0} has a unique quasiconformal extension f̃_{0} with Beltrami coefficient μ_{0}. Let h̃(λ) denote the Beltrami coefficient of the quasiconformal map F̃_{λ}∘f̃_{0}. By virtue of Theorem C (3), we then see that the map h̃ : Δ → M_{1} is holomorphic and satisfies Φ ∘ h̃ = h and h̃(0) = μ_{0}. Thus we have proved the following:

Proposition 2.2 ([11]). Suppose that a holomorphic map h : Δ → T (Δ) and a point μ_{0} ∈ M_{1} such that Φ(μ_{0}) = h(0) are given. Then, there exists a holomorphic map h̃ : Δ → M_{1} such that Φ ∘ h̃ = h on Δ and that h̃(0) = μ_{0}.

We will call h̃ a lift of h with h̃(0) = μ_{0}. This need not be uniquely determined by only the condition h̃(0) = μ_{0}.

Example 2.1. As the simplest example, we consider the holomorphic motion F (z, t) = f(tz)/t of the unit disk for a normalized univalent function f : Δ → ℂ so that f(0) = 0 and f′(0) = 1. (Of course, we set F (z, 0) = lim_{t→0} f(tz)/t = z.) By Theorem D, F extends to a holomorphic motion F̃ of the whole sphere. We then have a |λ|-quasiconformal extension F̃_{λ} of F_{λ}. Note that F̃_{λ} can be chosen so that F̃_{λ}(∞) = ∞. But, without this restriction, it turns out that F_{λ} can be extended to a |λ|^{2}-quasiconformal map (Krushkal’ [14]). In fact, the map h : Δ → T (Δ) defined by h(λ)(z) = S_{F_{λ}}(z) = λ^{2}S_{F} (λz) satisfies that h′(0) = 0. Thus the claim follows from the next theorem. Theorem E (Krushkal’ [14]). Suppose that a holomorphic map h : Δ → T (Δ) satisfies that h′(0) = ⋯ = h^{(m)}(0) = 0. Then d_{T} (h(0), h(t)) ≤ d_{Δ}(0, t^{m+1}) holds for every t ∈ Δ.

Problem. For the above h : Δ → T (Δ), can we choose a lift h̃ : Δ → M_{1} of h so that h̃′(0) = ⋯ = h̃^{(m)}(0) = 0?

If this is true, we have an easy proof of Theorem E by applying Schwarz’s lemma to the function t^{−m−1}(h̃(t) − μ_{0})/(1 − μ̄_{0}h̃(t)), where μ_{0} = h̃(0).

3. Univalence criteria and quasiconformal extensions

In this section, we explain how univalence criteria generate quasiconformal extension criteria by means of the λ-lemma. First, we state a general principle which leads to quasiconformal extensions of univalent functions.

Suppose that given are a hyperbolic simply connected domain U and an operation P on meromorphic functions on a plane domain D associated with a univalence criterion. Let N (D) be a class of meromorphic functions on D normalized at a point a ∈ D so that the operator P : N (D) → M(D) is injective, where M(D) is another class of meromorphic functions on D satisfyng the condition A at the point a, where the condition A is empty or φ(a) = p_{0}, where p_{0} is the point appearing just below. Further assume the following conditions:

(1) (univalence criterion). if f ∈ N (D) satisfies P_{f}(D) ⊂ U, the function f must be univalent in D, (2) id_{D} ∈ N (D) and P_{id_{D}}(z) = p_{0} for all z ∈ D, where p_{0} is a point in U,

(3) Hol_{0}(D, U) ⊂ P (N (D)), where Hol_{0}(D, U) is the class of holomorphic functions f on D with values in U satisfying the condition A,

(4) (holomorphic dependence of P). for a holomorphic family φ_{λ} in Hol_{0}(D, U) (i.e., the map λ ↦ φ_{λ}(z) is a ℂ̂-valued holomorphic function for every fixed z ∈ D), the corresponding functions f_{λ} in N (D) such that P_{f_{λ}} = φ_{λ} form a holomorphic family, too.

Under these circumstances, we can show the following claim.

Theorem 3.1. Let L : Δ → U be a Riemann mapping function of U with L(0) = p_{0}. If f ∈ N (D) satisfies P_{f}(D) ⊂ L(B_{k}) for some k ∈ [0, 1), the function f can be extended to a k-quasiconformal automorphism f̃ of ℂ̂, where B_{k} = {z ∈ ℂ; |z| ≤ k}. Further such an f̃ can be chosen so that f̃(∞) = ∞ when Hol_{0}(D, U) ⊂ P (N (D) ∩ Hol(D, ℂ)).

Proof. We define a holomorphic family φ_{λ} in Hol_{0}(D, U) parametrized through λ ∈ Δ by

φ_{λ}(z) = L (\frac{λ}{k}L^{−1} ∘ P_{f}(z)) .

Let F_{λ} be the holomorphic family in N (D) uniquely determined by P_{F_{λ}} = φ_{λ}. By the univalence criterion, we then observe that F (z, λ) = F_{λ}(z) is a holomorphic motion of D. When Hol_{0}(D, U) ⊂ P (N (D) ∩ Hol(D, ℂ)), we see D ⊂ ℂ since P_{id} = p_{0} ∈ Hol_{0}(D, U) implies id_{D} ∈ Hol(D, ℂ), therefore we can extend F to a holomorphic motion of D ∪ {∞} by setting F (∞, λ) = ∞. Now the optimal λ-lemma (Theorem D) produces a holomorphic motion F̃ of ℂ̂ whose restriction to D (or D ∪ {∞}) coincides with F. By Theorem C, each F̃_{λ} is a |λ|-quasiconformal extension of F_{λ}. Since P_{F_{k}} = φ_{k} = P_{f}, we have f = F_{k}, thus the proof is now complete.

Remarks. 1. In the above proof, the condition A was needed only for guaranteeing that L(k^{−1}λL^{−1} ∘ φ) ∈ P (N (D)) for all φ ∈ M(D) with φ(Δ) ⊂ B_{k} and λ ∈ Δ.

2. One may think that the above theorem can be generalized as follows: If P_{f}(D) ⊂ g_{k}(Δ) the function f ∈ N (D) can be extended to a k-quasiconformal automorphism of ℂ̂, where g(z, λ) = g_{λ}(z) is a holomorphic function from Δ × Δ into U satisfying that g_{λ} is univalent for each λ ∈ Δ \ {0}, while g_{0} is the constant function p_{0}. Actually, this can be shown by taking φ_{λ} = g_{λ} ∘ g_{k}^{−1} ∘ P_{f} however it is not a proper generalization because it is always true that g_{k}(Δ) ⊂ L(B_{k}). The last assertion is easily obtained by applying Schwarz’s lemma to the function λ ↦ L^{−1}(g(z, λ)).

Corollary 3.2. Under the same hypothesis as in Theorem 3.1, the norm of the Schwarzian derivative of the function f is estimated as ∥S_{f}∥_{2,D} ≤ 12k. When D is a unit disk Δ we have a better one: ∥S_{f}∥_{2,Δ} ≤ 6k. Furthermore if Hol_{0}(Δ, U) ⊂ P (N (Δ) ∩ Hol(Δ, ℂ)), we also have ∥T_{f}∥_{1,Δ} ≤ 6k.

Proof. By a theorem of Beardon and Gehring [5], we know that ∥S_{f}∥_{2,D} ≤ 12 for any univalent function f on the hyperbolic plane donain D. Hence our estimate follows from Lehto’s majorant principle. The other cases can be treated in the same fashion.

In the case D = Δ, we can consider the map h : Δ → T (Δ) defined by h(λ) = S_{F_{λ}}, where F_{λ} is the holomorphic family constructed above. It is easy to see that h : Δ → T (Δ) is actually holomorphic and h(0) = 0. If h′(0) = 0, by Theorem E, we would then have a better estimate d_{T} (h(λ), 0) ≤ d_{Δ}(0, λ^{2}), which implies that F_{λ} can be extended to a |λ|^{2}-quasiconformal automorphism of ℂ̂. We will see below when h′(0) = 0 happens in concrete examples.

4. Applications

Now we apply the theorem shown in the preceeding section to several examples.

First we consider the operation P_{f} = f′ for non-constant holomorphic functions on a convex domain D ⊂ ℂ. Now fix a point a ∈ D, and consider the class N (D) = {f ∈ Hol(D, ℂ); f(a) = 0} and M(D) = Hol(D, ℂ). (Therefore, the condition A is vacuous here.) For β ∈ (−π/2, π/2), we set U_{β} = {z ∈ ℂ^{∗}; | arg z − β| < π/2}. Then the Noshiro-Warschawski theorem yields that if P_{f}(D) ⊂ U_{β} the function f ∈ N (D) should be univalent. Hence Theorem 3.1 is now applicable. The Riemann mapping function L_{β} : Δ → U_{β} of U_{β} with L_{β}(0) = 1 is given by L_{β}(z) = (1 + e^{2iβ}z)/(1 − z). Since the assumption f(a) = 0 is not essential for quasiconformal extendability, we have the next

Theorem 4.1. Let D be a convex domain and take a point a in D. Suppose that a holomorphic function f on D satisfies f′(D) ⊂ L_{β}(B_{k}). Then the function f can be extended to a k-quasiconformal automorphism of the Riemann sphere fixing ∞.

Note that

L_{β}(B_{k}) = \{z ∈ ℂ; |z − \frac{1 + e^{2iβ}k^{2}}{1 − k^{2}} | ≤ \frac{2k cos β}{1 − k^{2}} \} .

Suppose that the convex domain D is also a quasidisk, in other words, the Riemann mapping function g : Δ → D of D can be extended to a quasiconformal mapping. Under the same notation as in Section 3, we consider h(λ) = S_{F_{λ}∘g} = g^{∗}S_{F_{λ}} + S_{g}, where g^{∗} : B_{2}(D) → B_{2}(Δ) is the isometric isomorphism defined by the pullback g^{∗}φ(z) = φ(g(z))g′(z)^{2} by g. We set ψ = k^{−1}L_{β}^{−1} ∘ f′. We then have φ_{λ} = L_{β}(λψ) = 1 + (1 + e^{2iβ})λψ + O(λ^{2}) as λ → 0. Hence, h(λ) = g^{∗}(φ′′_{λ}/φ_{λ} − 3(φ′_{λ}/φ_{λ})^{2}/2) + S_{g} = S_{g} + (1 + e^{2iβ})λg^{∗}(ψ′′) + O(λ^{2}) as λ → 0. Theorefore, we can see that h′(0) = (1 + e^{2iβ})g^{∗}(ψ′′). In particular, h′(0) = 0 if and only if ψ′′ = 0, i.e., f′(z) = L_{β}(c(z + d)) with constants c and d satisfying sup_{w∈∂D} |w + d| ≤ k/|c|, and in that case f can be extended to a k^{2}-quasiconformal automorphism of ℂ̂ by Theorem E.

It is not clear from the proof whether Theorem 4.1 is best possible or not. In the case that β = 0 and D = Δ, actually, this is best possible. In fact, take the normalized holomorphic function f_{2} : Δ → ℂ determined by the relation f′_{2}(z) = L_{0}(kz^{2}) = 1 − 2kz^{2} + …. Explicitly,

f_{2}(z) = −z + \frac{1}{\sqrt{k}} log \frac{1 + \sqrt{k}z}{1 − \sqrt{k}z} .

Then the meromorphic function g_{2}(z) = 1/f_{2}(1/z) belongs to the classical class Σ_{0} of normalized univalent meromorphic function on |z| > 1 and can be represented by the power series

g_{2}(z) = z − \frac{2k}{3} z^{−1} + \frac{2k^{2}}{45} z^{−3} + \frac{46k^{3}}{945} z^{−5} + ⋯

in |z| > 1. On the other hand, by a theorem of Schiffer (see [10, §4.7]), any g(z) = z + b_{1}/z + b_{2}/z^{2} + … in Σ_{0} satisfies |b_{2}| ≤ 2/3. Now, from the Lehto majorant principle [17, §§3.5, 3.6], it follows that a function g ∈ Σ_{0} admitting κ-quasiconformal mapping of ℂ̂ should satisfy |b_{2}| ≤ 2κ/3. Hence, f_{2} cannot be extended to any κ-quasiconformal mapping for an arbitrary κ < k.

A normalized analytic function f on the unit disk is called close-to-convex if

Ref′(z)/g′(z) > 0 ∀z ∈ Δ

for some univalent function g on Δ whose image is convex. Note that the function g need not be normalized here. For subclasses of close-to-convex functions, the following statement is immediately verified from Theorem 4.1.

Corollary 4.2. Let g be a conformal mapping from the unit disk onto a convex domain D which admits a k_{1}-quasiconformal automorphism of the Riemann sphere. For constants k_{2} ∈ [0, 1) and β ∈ (−π/2, π/2), suppose that an analytic function f on the unit disk satisfies that f′(z)/g′(z) ∈ L_{β}(B_{k_{2}}) for any z ∈ Δ. Then f can be extended to a (k_{1} + k_{2})/(1 + k_{1}k_{2})-quasiconformal automorphism of the Riemann sphere.

Proof. The function h(z) = f(g^{−1}(z)) satisfies h′(D) ⊂ L_{β}(B_{k_{2}}). Now the result follows from Theorem 4.1 and from the fact that the composition of k_{1} and k_{2}-quasiconformal mappings becomes a (k_{1} + k_{2})/(1 + k_{1}k_{2})-quasiconformal mapping.

Remark. The λ-lemma does not tell us any explicit construction of the quasiconformal extension. At least, in this case, we have a concrete way of k′-quasiconformal extension of a function f ∈ N (Δ) with f′(Δ) ⊂ L_{0}(B_{k}), where k′ = 2k/(1 + k^{2}). In fact, writing as f(z) = (1+k^{2})z/(1−k^{2})+p(z), we can extend f by setting f(z) = (1+k^{2})z/(1−k^{2})+p(1/z̄) outside the unit disk Δ.

Secondly, we consider the operation P_{f}(z) = zf′(z)/f(z) on holomorphic functions f on the unit disk Δ. Let N (Δ) be the set of holomorphic functions f on the unit disk normalized so that f(0) = f′(0) − 1 = 0 and M(Δ) the set of holomorphic functions φ on Δ satisfying the condition A: φ(0) = 1. Fix β ∈ (−π/2, π/2). Then the condition P_{f}(Δ) ⊂ U_{β} says that f is β-spirallike, in particular, univalent on Δ (see, for example, [10]). For a φ ∈ M(D), we have the following expression of the function f ∈ N (Δ) with P_{f} = φ :

f(z) = z exp (∫_{0}^{z} \frac{φ(ζ) − 1}{ζ}dζ) .

In particular, the holomorphic dependence of the operator P is evident. So, Theorem 3.1 is now applicable.

Theorem 4.3. Let β ∈ (−π/2, π/2) and f be a holomorphic function on the unit disk with f′(0) = 1. Suppose that f satisfies zf′(z)/f(z) ∈ L_{β}(B_{k}) on Δ for some constant k ∈ [0, 1). Then the function f can be extended to a k-quasiconformal automorphism of the Riemann sphere fixing ∞. In particular, we have ∥T_{f}∥_{1,Δ} ≤ 6k and ∥S_{f}∥_{2,Δ} ≤ 6k.

When β = 0 (i.e., the case of ordinary starlike functions), the above result is best possible. In fact, the function f ∈ N (Δ) satisfying P_{f}(z) = L_{0}(kz^{2}) = (1+kz^{2})/(1−kz^{2}) has the form z/(1 − kz^{2}), therefore has the k-quasiconformal extension by setting f(z) = z/(1 − kz/z̄) outside Δ, which turns to be conjugate to the affine map z ↦ z − kz̄ by the inversion z ↦ 1/z, thus to be extremal.

On the other hand, for general β ∈ (−π/2, π/2), it is unknown whether the above result is best possible or not. At least we know so far that the function f ∈ N (Δ) determined by f′(z) = L_{β}(kz^{2}) with β ≠ 0 has a k′-quasiconformal extension with some k′ < k, which we show now. The function f has the form z(1 − kz^{2})^{−α}, where α = (1 + e^{2iβ})/2. Then the function F (z) = 1/f(1/z) = z(1 − k/z^{2})^{α} on Δ^{∗} can be extended to a quasiconformal mapping by setting F (z) = z(1 − kz̄^{2}|z|^{−2/α})^{α} on Δ. The Beltrami coefficient μ of F has the form μ(z) = −ke^{2iβ}|z|^{i tan β} on Δ, in particular, |μ(z)| = k a.e. on Δ thus F is a k-quasiconformal automorphism of ℂ̂. This mapping F is not extremal. In fact, for any integrable holomorphic quadratic differential φ(z)dz^{2} on the unit disk Δ, we have

∫∫ _{Δ} μ(z)φ(z)dxdy = ∫_{0}^{1} ∫_{0}^{2π} μ(r)φ(re^{iθ})rdθdr

= 2πφ(0) ∫_{0}^{1} rμ(r)dr = \frac{−2πke^{2iβ}φ(0)}{2 + i tan β}.

From the sharp estimate π|φ(0)| ≤ ∥φ∥ = ∫∫_{Δ} |φ(z)|dxdy, it then follows that

sup_{∥φ∥≤1} |∫∫ _{Δ} μ(z)φ(z)dxdy| = \frac{2k}{|2 + i tan β|} < k = ∥μ∥_{∞}

if β ≠ 0. Hence F does not satisfy the Hamilton-Krushkal’ condition, equivalently, F is not extremal.

Now we examine the condition for h′(0) = 0, where h(λ) = S_{F_{λ}} and F_{λ} is as in Section 3. Letting ψ = k^{−1}L_{β}^{−1} ∘ P_{f}, we see φ_{λ} = L_{β}(λψ) = 1 + (1 + e^{2iβ})λψ + O(λ^{2}) as λ → 0. A straightforward calculation shows that

z^{2}S_{f}(z) = z (\frac{zφ′(z)}{φ(z)})′ − \frac{1}{2} (\frac{zφ′(z)}{φ(z)})^{2} + \frac{1}{2}(1 − φ(z)^{2})

if a non-constant holomorphic function f sastisfies zf′(z)/f(z) = φ(z) (cf. [23]). Thus we have z^{2}S_{F_{λ}}(z) = (1 + e^{2iβ})λ(z(zψ′(z))′ − ψ(z)) + O(λ^{2}), which implies

h′(0)(z) = (1 + e^{2iβ})z^{−2}(z^{2}ψ′′(z) + zψ′(z) − ψ(z)).

In particular, h′(0) = 0 if and only if the function ψ is a holomorphic solution of the Fuchsian differential equation z^{2}ψ′′(z) + zψ′(z) − ψ(z) = 0. The indicial equation ρ(ρ − 1) + ρ − 1 = ρ^{2} − 1 = 0 has the roots 1 and −1, thus a holomorphic solution near the origin is a constant multiple of the fundamental solution ψ(z) = z corresponding to the root 1. Therefore P_{f}(z) = L_{β}(cz), where c is a constant with |c| ≤ k. By Theorem E, we conclude that the function f has a k^{2}-quasiconformal extension to the sphere if f satisfies zf′(z)/f(z) = P_{β}(kz). Such an f is nothing but the function z/(1 − kz)^{1+exp(2iβ)}. The function z/(1 − z)^{1+exp(2iβ)} is sometimes called the β-spirallike Koebe function.

Remark. Here we mention a relation with strongly starlike functions. A normalized holomorphic function f on the unit disk is called strongly starlike of order α if | arg P_{f}(z)| = | arg(zf′(z)/f(z))| ≤ πα/2 on Δ, where α is a constant with 0 ≤ α ≤ 1. In [12], it is shown that such a function can be extended to a sin(πα/2)-quasiconformal automorphism of ℂ̂ fixing ∞ by an explicit way. In particular, we have ∥S_{f}∥_{2,Δ} ≤ 6 sin(πα/2). There is a problem: Can one extend a strongly starlike function of order α to an αquasiconformal automorphism of ℂ̂? If this is true, we should have ∥S_{f}∥_{2,Δ} ≤ 6α. So far, the author only knows that |S_{f}(0)| ≤ 6α for a strongly starlike function f of order α and the equality holds if and only if f is a rotation of the function F determined by the relation zF′(z)/F (z) = {(1 + z^{2})/(1 − z^{2})}^{α}. This fact can be shown in the same fashion as in [8]. (See also [23].)

On the other hand, an elementary geomerty shows that a function f ∈ N (Δ) satisfying P_{f}(Δ) ⊂ L_{0}(B_{k}) is strongly starlike of order α, where α is the number determined by sin(πα/2) = 2k/(1 + k^{2}), equivalently, tan(πα/4) = k. Because πα/4 < k < α(< sin(πα/2)) for 0 < k < 1, our result supports the affirmative answer to the problem above.

Next we consider the operation P_{f}(z) = 1 + zf′′(z)/f′(z). Let N (Δ) be the set of holomorphic functions f on the unit disk normalized so that f(0) = f′(0) − 1 = 0 and M(Δ) the set of holomorphic functions φ on Δ with the condition A: φ(0) = 1. Then the condition P_{f}(Δ) ⊂ U_{0} = {z ∈ ℂ; Rez > 0} means that the function f is convex, in particular, univalent in Δ. Letting g(z) = zf′(z), we have, by the Alexander theorem, P_{f}(z) = zg′(z)/g(z). Since f is recovered by the formula f(z) = ∫_{0}^{z} g(ζ)/ζdζ from the function g, in combination with the last example, we obtain the holomorphic dependence of the operation P. Therefore, by virtue of Theorem 3.1, we have the following:

Theorem 4.4. Suppose that a non-constant holomorphic function f on the unit disk Δ satisfies 1 + zf′′(z)/f′(z) ∈ L_{0}(B_{k}) on Δ for some constant k ∈ [0, 1). Then f can be extended to a k-quasiconformal automorphism of the Riemann sphere fixing ∞.

Corollary 4.5. Under the same situation in the above theorem, we have the norm estimate of the Schwarzian derivative: ∥S_{f}∥_{2,Δ} ≤ 2k.

This follows from the fact ∥S_{f}∥_{2,Δ} ≤ 2 for any convex function f on Δ (see [20] or [16]) and the Lehto majorant principle. We remark that this corollary also implies kquasiconformal extendability of such a function f as above by the Ahlfors-Weill theorem [1].

Finally, we consider the derivative of h(λ) = S_{F_{λ}}. We set ψ = k^{−1}L_{0}^{−1} ∘ P_{f}. Then φ_{λ} = L_{0}(λψ) = 1 + 2λψ + O(λ^{2}) as λ → 0. In this case, z^{2}S_{F_{λ}}(z) = zφ′_{λ}(z) − (φ_{λ}(z)^{2} − 1)/2 = 2λ(zψ′(z) − ψ(z)) + O(λ^{2}), in particular, h′(0)(z) = 2z^{−2}(zψ′(z) − ψ(z)). Hence, h′(0) = 0 if and only if ψ(z) = cz, where c is a constant with |c| ≤ 1. But, the function f such that P_{f}(z) = L_{0}(kz) turns out to be the Möbius transformation f(z) = z/(1 − kz).

We conclude this section by giving an example of another type. Consider here the Schwarzian derivative as the operation P, i.e., P_{f} = S_{f}. Let D be a hyperbolic simply connected plane domain and take a finite point a ∈ D. And let N (D) the set of normalized locally univalent meromorphic functions f on D so that f(a) = f′(a) − 1 = f′′(a) = 0 and M(D) the set of all holomorphic functions φ on D such that φ(z) = O(z^{−4}) as z → ∞ if ∞ ∈ D. In this case, the condition A is vacuous. Now we restrict our attention to the case that D = Δ with a = 0. Then, by Nehari’s result [19], the condition |S_{f}(z)| ≤ π^{2}/2 on the unit disk Δ forces f ∈ N (Δ) to be univalent. The holomorphic dependence of the Schwarzian derivative operator is well-known in the complex analytic theory of Teichmüller spaces, so we can apply Theorem 3.1 to show the following.

Theorem 4.6. Let k be a constant in [0, 1). If a function f ∈ N (Δ) satisfies that |S_{f}(z)| ≤ π^{2}k/2 on Δ, then f can be extended to a k-quasiconformal automorphism of the whole sphere.

This resut is, of course, not new. And we remark that the above method is also applicable for any quasidisk D.

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Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan

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