Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains
aa r X i v : . [ m a t h . C V ] S e p CRITERIA FOR UNIVALENCE AND QUASICONFORMALEXTENSION FOR HARMONIC MAPPINGS ON PLANARDOMAINS
IASON EFRAIMIDIS
Abstract.
If Ω is a simply connected domain in C then, according to theAhlfors-Gehring theorem, Ω is a quasidisk if and only if there exists a sufficientcondition for the univalence of holomorphic functions in Ω in relation to thegrowth of their Schwarzian derivative. We extend this theorem to harmonicmappings by proving a univalence criterion on quasidisks. We also show thatthe mappings satisfying this criterion admit a homeomorphic extension to C and, under the additional assumption of quasiconformality in Ω, they admita quasiconformal extension to C .The Ahlfors-Gehring theorem has been extended to finitely connected do-mains Ω by Osgood, Beardon and Gehring, who showed that a Schwarziancriterion for univalence holds in Ω if and only if the components of ∂ Ω are ei-ther points or quasicircles. We generalize this theorem to harmonic mappings. Introduction
Schwarzian derivative.
For a locally univalent analytic function f theSchwarzian derivative is defined by Sf = (cid:0) f ′′ /f ′ (cid:1) ′ − (cid:0) f ′′ /f ′ (cid:1) . This operator vanishes identically if and only if f is a M¨obius transformation.According to a classical theorem of Nehari the bound | Sf ( z ) | ≤ t (1 − | z | ) , z ∈ D , (1)for t = 1 implies the global univalence of f in the unit disk D , while anotherclassical result, proved by Ahlfors and Weill, gives an explicit quasiconformalextension of f to C under the assumption that f satisfies (1) with t < C , meaning that it has at least three boundarypoints, and let π : D → Ω be a universal covering map. Then the density λ Ω ofthe hyperbolic (Poincar´e) metric in Ω is defined by λ Ω (cid:0) π ( z ) (cid:1) | π ′ ( z ) | = λ D ( z ) = 11 − | z | , z ∈ D , Mathematics Subject Classification.
Key words and phrases. harmonic mappings, Schwarzian derivative, univalence criterion,quasiconformal extension. and is independent of the choice for the covering π . The size of the Schwarzianderivative of a locally univalent holomorphic function f : Ω → C is measured byits norm, given by k Sf k Ω = sup z ∈ Ω λ Ω ( z ) − | Sf ( z ) | . (2)The inner radius of Ω is defined as the number σ (Ω) = sup { c ≥ k Sf k Ω ≤ c ⇒ f univalent } . This domain constant is M¨obius invariant. We say that a univalence criterionholds in Ω if and only if σ (Ω) >
0. We have that σ ( D ) = 2 since, as shown byHille, the constant 2 in Nehari’s theorem is sharp. Lehtinen showed that everysimply connected domain Ω satisfies σ (Ω) ≤
2, with equality only in the casewhen Ω is a disk or a half-plane. See Lehto’s book [14, Ch.III, §
5] for moreinformation on the inner radius.A domain Ω is a quasidisk if it is the image of D under a quasiconformalself-map of C . The boundary of a quasidisk is called a quasicircle.Our starting point is the following theorem by Ahlfors and Gehring. Theorem A ([1, 7]) . Let Ω be a simply connected domain in C . Then σ (Ω) > if and only if Ω is a quasidisk. Moreover, any holomorphic function f : Ω → C that satisfies k Sf k Ω < σ (Ω) admits a quasiconformal extension to C . Ahlfors [1] proved the quasiconformal extension criterion on a quasidisk statedhere; the univalence criterion follows from it. The fact that such a criterion canonly occur on a quasidisk was shown by Gehring [7]. This has been generalizedto a class of multiply connected domains with the following theorem.
Theorem B ([2, 18]) . Let Ω be a finitely connected domain in C . Then σ (Ω) > if and only if every boundary component of Ω is either a point or a quasicircle. The univalence criterion was proved by Osgood [18], while the fact that theclass of domains where it holds cannot be enlarged was proved by Beardon andGehring [2]. We note that for finitely connected domains the property of ∂ Ωdescribed in this theorem characterizes the class of uniform domains, introducedby Martio and Sarvas [16] (see also [8, § Harmonic mappings.
A complex-valued harmonic mapping f in a simplyconnected domain Ω ⊂ C has a canonical decomposition f = h + g , where h and g are analytic in Ω. The mapping f is locally univalent if and only if its Jacobian J f = | h ′ | − | g ′ | does not vanish, and is said to be orientation-preserving ifits dilatation ω = g ′ /h ′ satisfies | ω | < f is normalized if h ( z ) = g ( z ) = 0 and h ′ ( z ) = 1 for some specified z ∈ Ω.The Schwarzian derivative has been extended to harmonic mappings by twocomplementary definitions: a first one appeared in [4] and another was later
RITERIA FOR UNIVALENCE AND QUASICONFORMAL EXTENSION 3 introduced by Hern´andez and Mart´ın in [11]. We will follow the latter, whichseems to be better suited when one does not wish to consider the Weierstarss-Enneper lift to a minimal surface. Hence, the Schwarzian derivative of a locallyunivalent harmonic mapping f is defined by S f = ρ zz − ( ρ z ) , ρ = log J f , where J f = | f z | − | f z | is the Jacobian. If Ω is simply connected and, therefore,the decomposition f = h + g is valid, then the above takes the form S f = Sh + ω − | ω | (cid:18) h ′′ h ′ ω ′ − ω ′′ (cid:19) − (cid:18) ω ′ ω − | ω | (cid:19) . (3)Note that we are using the notation Sf when we know that the mapping f isholomorphic and the notation S f , with the mapping f as a subscript, in the moregeneral setting of harmonic mappings. This operator satisfies the chain rule, forif ϕ is analytic in Ω and such that the composition f ◦ ϕ is well defined then wehave that S f ◦ ϕ = S f ◦ ϕ ( ϕ ′ ) + Sϕ. (4)The Schwarzian norm of a harmonic mapping f is defined exactly as in (2). Itwas shown in [11] that k S f k Ω = 0 implies that f is an affine map of a M¨obiustransformation and is, therefore, univalent.1.3. Main results.
We define the harmonic inner radius of a hyperbolic domainΩ in C as the constant σ H (Ω) = sup { t ≥ f harmonic in Ω with k S f k Ω ≤ t ⇒ f univalent } . Evidently, σ H (Ω) ≤ σ (Ω) , (5)since every holomorphic function is harmonic. This shows that if σ H (Ω) > D it was shown in [12] that σ H ( D ) > Theorem 1.
Let Ω be a quasidisk. Then there exists a constant c > , dependingonly on σ (Ω) , such that if f is harmonic in Ω with k S f k Ω ≤ c then f is univalentin Ω and admits a homeomorphic extension to C . The proof of the univalence criterion in Theorem 1 follows closely the reasoningin [12]: We show that if a harmonic mapping f = h + g has small Schwarzianderivative then so does its analytic part h , and therefore h is univalent by Theo-rem A. The same can then be said about h + ag , the analytic part of the affinetransformation f + af , a ∈ D . Finally, Hurwitz’ theorem shows that h + ag isunivalent for every a ∈ D and by an elementary rotational argument we get that f is injective. The crucial step in the generalization from D to a quasidisk involvesthe hyperbolic derivative of admissible dilatations and is given in Lemma 5. Thehomeomorphic extension under these hypotheses is novel even for the unit disk.Further, for more general domains we prove the following theorem. I. EFRAIMIDIS
Theorem 2.
Let Ω be a finitely connected domain. The following are equivalent. (i) Every boundary component of Ω is either a point or a quasicircle. (ii) σ (Ω) > σ H (Ω) > ⇒ (ii) follows trivially from (5). We prove the direction (i) ⇒ (iii) in Section 6by using Osgood’s [18] quasiconformal decomposition of Ω and the homeomorphicextension of Theorem 1.Finally, we give sufficient conditions for a harmonic mapping defined on aquasidisk to admit a quasiconformal extension to C . Theorem 3.
Let Ω be a quasidisk and let d ∈ [0 , . Then there exists a con-stant c > , depending only on d and on σ (Ω) , such that if f is harmonic in Ω with k S f k Ω ≤ c and its dilatation satisfies sup z ∈ Ω | ω ( z ) | ≤ d then f admits aquasiconformal extension to C . If d = 0 then f is analytic and we recover Ahlfors’ [1] theorem. For our proof ofTheorem 3 we consider the dilation Ω r ( i.e. the image of | z | < r , for r <
1, underthe Riemman mapping of Ω) and use quasiconformal reflections to obtain a K -quasiconformal extension of f (cid:12)(cid:12) Ω r to C . The desired extension is then harvestedas the limit for r →
1, once we prove that K is independent of r by studyingthe cross-ratio of points on the image of ∂ Ω r under f . For the latter we use theinsightful ideas of [10]. We prove Theorem 3 in Section 5.We note that for the case when Ω is the unit disk D , Theorem 3 was provedin [12]. However, the slightly stronger statement made there, namely that theconstants c and d in the hypotheses are independent, does not seem to followfrom the suggestion of the authors to argue as in [10] for a proof. It is interestingto ask if this stronger statement can be rigorously proved.2. Preliminaries
Bounded Schwarzian derivative.
A well-known theorem of Pommerenke[19] states that if ϕ is analytic and locally univalent in D then(1 − | z | ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ ′′ ( z ) ϕ ′ ( z ) − z − | z | (cid:12)(cid:12)(cid:12)(cid:12) ≤ r k Sϕ k D , z ∈ D . (6)From this, a simple application of Montel’s theorem shows that the set of func-tions { ϕ : k Sϕ k D ≤ c } , where c >
0, constitutes a normal family (the set of fucntions (log ϕ ′ ) ′ is locallyuniformly bounded in D , and so is the set of functions ϕ ).According to Theorem 6 in [11], if f = h + g is harmonic and locally univalentin D then k S f k D < ∞ if and only if k Sh k D < ∞ . (7) RITERIA FOR UNIVALENCE AND QUASICONFORMAL EXTENSION 5
Nomralizations for harmonic mappings.
Let Ω be a simply connecteddomain that contains the origin and let t ≥
0. We then denote by F t (Ω) the setof all sense-preserving harmonic mappings f = h + g in Ω which satisfy k S f k Ω ≤ t and are normalized by h (0) = g (0) = 0 and h ′ (0) = 1. Let F t (Ω) = { f ∈ F t (Ω) : g ′ (0) = 0 } . The following proposition is well known among experts but we will include aproof here for the convenience of the reader. We note that the compactness of F t ( D ) was shown in [5]. Proposition 1.
The family F t (Ω) is normal. The family F t (Ω) is normal andcompact.Proof. To show that F t ( D ) is normal we observe that the set { h : f = h + g ∈ F t ( D ) } is a normal family in view of inequality (6) and since k Sh k D is bounded by(7) (even more so, a close inspection of [11, Thm.6] shows that it is uniformlybounded). Also, again by (6), the functions h ′ are locally uniformly bounded in D and, in view of the condition | g ′ | < | h ′ | , so are the functions g ′ . Hence thefamily { g : f ∈ F t ( D ) } is also normal. Thus F t ( D ) is a normal family.For a simply connected domain Ω, with 0 ∈ Ω, let f ∈ F t (Ω) and consider themapping F = ϕ ′ (0) − f ◦ ϕ , where ϕ is a Riemann map for which Ω = ϕ ( D ) and ϕ (0) = 0. By the chain rule (4) we have that S F ( z ) = S f (cid:0) ϕ ( z ) (cid:1) ϕ ′ ( z ) + Sϕ ( z ) , z ∈ D . We then compute | S F ( z ) − Sϕ ( z ) | λ D ( z ) = | S f (cid:0) ϕ ( z ) (cid:1) | λ Ω (cid:0) ϕ ( z ) (cid:1) , z ∈ D , (8)which shows that k S F − Sϕ k D = k S f k Ω . According to Kraus’ theorem [14, Ch.II,Thm.1.3] we have that k Sϕ k D ≤
6. Therefore, we get that k S F k D ≤ k S f k Ω + k Sϕ k D ≤ t + 6 . Hence the set { ϕ ′ (0) − f ◦ ϕ : f ∈ F t (Ω) } is included in the normal family F t +6 ( D )and is, therefore, a normal family itself. The claim that F t (Ω) is normal followsdirectly from this.Any f ∈ F t (Ω) can be written as an affine transform of a mapping in F t (Ω),in particular, if b = g ′ (0) we may write f = f + b f for some f ∈ F t (Ω).Hence | f | ≤ | f | and with the observation that Montel’s criterion for normalityremains valid for families of harmonic mappings (see [6, p.80]) we conclude that F t (Ω) is normal.Finally, the compactness of F t (Ω), as shown in the course of the proof ofTheorem 3 in [5], amounts to the observation that if f n is a sequence in F t (Ω)that converges to f locally uniformly in Ω then S f n → S f pointwise in Ω. Hence f ∈ F t (Ω) and so this class is compact. (cid:3) I. EFRAIMIDIS
Affine invariance.
Let a ∈ D and consider the affine transformation of amapping f in F t (Ω) given by F ( z ) = A a f ( z ) = f ( z ) + a f ( z )1 + ag ′ (0) , z ∈ Ω . (9)Now F ∈ F t (Ω) since it satisfies S F ≡ S f (see [11, Prop.1]) and the correspond-ing normalizations. It can easily be seen that the dilatation of F is given by ω F = νϕ a ◦ ω , where ν ∈ T (= ∂ D ) and ϕ a ( z ) = a + z az , z ∈ D . If we make the choice a = − ω (0) then F will have the additional normalization ω F (0) = νϕ a ( − a ) = 0 and will therefore belong to F t (Ω).2.4. Quasiconformal mappings.
A sense-preserving homeomorphism f : Ω → C is said to be K -quasiconformal, K ≥
1, if it is absolutely continuous on linesand satisfies | f z | ≤ k | f z | , where k = ( K − / ( K + 1), almost everywhere in Ω.A mapping is called quasiconformal if it is K -quasiconformal for some K ≥
1. The1-quasiconformal mappings are the conformal mappings. Note that a harmonicmapping is quasiconformal if its dilatation satisfies | ω | ≤ k < K -quasidisk, and its boundary a K -quasicircle, if it isthe image of D under a K -quasiconformal self-map of C . According to a theoremof Ahlfors [1] a Jordan curve γ ⊂ C is a quasicircle if and only if for all points z j ∈ γ, j = 1 , , ,
4, such that z and z separate z and z , the cross-ratio( z , z , z , z ) = ( z − z )( z − z )( z − z )( z − z )satisfies | ( z , z , z , z ) | ≤ C, for some constant C > and Ω be the complementary components of a Jordan curve γ ⊂ C .Then a sense-reversing homeomorphism λ of the sphere onto itself is a reflectionacross γ if it maps Ω onto Ω and keeps every point on γ fixed. According toa theorem of K¨uhnau [13] (see also Theorem 2.1.4 in [8]) γ is a K -quasicircle ifand only if it admits a reflection λ such that λ ( z ) is K -quasiconformal.3. Preparatory lemmas
Adaptations to general domains.
The following proposition is a straight-forward generalization of (7).
Proposition 2.
Let Ω be a simply connected domain and f = h + g a sense-preserving locally univalent harmonic mapping in Ω . Then k S f k Ω < ∞ if andonly if k S h k Ω < ∞ .Proof. Let ϕ be a Riemann map for which Ω = ϕ ( D ) and consider the mappings F = f ◦ ϕ and H = h ◦ ϕ . By a calculation we saw in (8) we readily have that k S F − Sϕ k D = k S f k Ω and k SH − Sϕ k D = k Sh k Ω . RITERIA FOR UNIVALENCE AND QUASICONFORMAL EXTENSION 7
The proposition is a direct consequence of Theorem 6 in [11] and Kraus’ theorem k Sϕ k D ≤ (cid:3) We now generalize inequality (6) to an arbitrary hyperbolic domain Ω. Wewrite d ( z ) = dist ( z, ∂ Ω) for the distance of a point z in Ω to the boundary ∂ Ω. Proposition 3. If h is analytic and locally univalent in Ω then (cid:12)(cid:12)(cid:12)(cid:12) h ′′ ( z ) h ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ d ( z ) r k Sh k Ω , z ∈ Ω . Proof.
Let α ∈ Ω , d = d ( α ) and consider the disk ∆ = { z : | z − α | < d } . We write z = α + d ζ ∈ ∆, for ζ ∈ D , and note that d λ ∆ ( z ) = λ D ( ζ ). Since ∆ ⊂ Ω we haveby the comparison principle [3, Thm.8.1] that λ ∆ ( z ) ≥ λ Ω ( z ), for all z ∈ ∆. Let H ( ζ ) = h ( z ) and observe that SH ( ζ ) = d Sh ( z ). We have that k SH k D = sup ζ ∈ D | SH ( ζ ) | λ D ( ζ ) = sup z ∈ ∆ | Sh ( z ) | λ ∆ ( z ) ≤ sup z ∈ Ω | Sh ( z ) | λ Ω ( z ) = k Sh k Ω . We now apply inequality (6) to the function H and evaluate at ζ = 0 to obtain d ( α ) (cid:12)(cid:12)(cid:12)(cid:12) h ′′ ( α ) h ′ ( α ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) H ′′ (0) H ′ (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ r k SH k D ≤ r k Sh k Ω . The proof is complete. (cid:3)
The hyperbolic derivative. If ω : Ω → D is an analytic function then itshyperbolic derivative is given by ω ∗ ( z ) = ω ′ ( z ) λ Ω ( z ) (1 − | ω ( z ) | ) , z ∈ Ω , and the quantity k ω ∗ k = sup z ∈ Ω | ω ∗ ( z ) | is called the hyperbolic norm of ω . Inview of the generalized Schwarz-Pick lemma [3, Thm.10.5] we always have that k ω ∗ k ≤
1. The hyperbolic derivative satisfies the chain rule ( ω ◦ ϕ ) ∗ = ω ∗ ◦ ϕ · ϕ ∗ for any two functions for which the composition is well defined.It has been shown in [9] that for any analytic ω : D → D it holds that(1 − | z | ) | ω ′′ ( z ) | − | ω ( z ) | ≤ C k ω ∗ k , z ∈ D , (10)for some constant C >
0. We will now generalize this to an arbitrary hyperbolicdomain Ω. Again, here d ( z ) = dist ( z, ∂ Ω).
Proposition 4. If ω : Ω → D is analytic then d ( z ) | ω ′′ ( z ) | − | ω ( z ) | ≤ C k ω ∗ k , z ∈ Ω , for some constant C > . I. EFRAIMIDIS
Proof.
We proceed as in the proof of Proposition 3, by fixing α ∈ Ω and writing d = d ( α ) , ∆ = { z : | z − α | < d } and z = α + d ζ ∈ ∆, for ζ ∈ D . We set ψ ( ζ ) = ω ( z ) and compute k ψ ∗ k = sup ζ ∈ D | ψ ′ ( ζ ) | λ D ( ζ ) (1 − | ψ ( ζ ) | )= sup z ∈ ∆ | ω ′ ( z ) | λ ∆ ( z ) (1 − | ω ( z ) | ) ≤ sup z ∈ Ω | ω ′ ( z ) | λ Ω ( z ) (1 − | ω ( z ) | )= k ω ∗ k . The proof is completed by applying inequality (10) for ζ = 0 to the function ψ . (cid:3) Admissible dilatations.
Let Ω be a simply connected domain with 0 ∈ Ωand let A t (Ω) and A t (Ω) denote the classes of admissible dilatations for mappingsin F t (Ω) and F t (Ω), respectively. Let also R t (Ω) = max ω ∈A t (Ω) k ω ∗ k . Applying the affine transformation (9) to a mapping f ∈ F t (Ω), as we havealready seen, we can get a mapping F = A a f which, for an appropriate choice of a ∈ D , belongs to ∈ F t (Ω). The dilatation of F is ω F = νϕ a ◦ ω , for some ν ∈ T .A straightforward computation can show that | ω ∗ F | = | ω ∗ | , so that we have analternative expression for R t given by R t (Ω) = sup ω ∈A t (Ω) k ω ∗ k . (11)This was first observed in [5, Lem.1]. A compactness argument was used in [12]to show that R t ( D ) → t → + . Here we will prove the following. Lemma 5.
It holds that R t (Ω) ≤ R t ( D ) for all t > . An immediate consequence is that R t (Ω) → t → + , and this fact isan important ingredient in the proofs of Theorems 1 and 3. For the proof ofLemma 5 we need to recall the inequalities14 ≤ d ( z ) λ Ω ( z ) ≤ , z ∈ Ω , (12)where d ( z ) = dist ( z, ∂ Ω), which amount to Koebe’s 1/4-theorem and the com-parison principle; see [14, Ch.I, § Proof of Lemma 5.
Fix t > f = h + g in F t (Ω), with dilatation ω , for which R t (Ω) = k ω ∗ k . There exists a sequence ofpoints { z n } in Ω for which k ω ∗ k = lim n →∞ | ω ∗ ( z n ) | . Let r n = d ( z n ) and considerthe disks ∆ n = { z : | z − z n | < r n } . Hereafter we use the notation ζ ∈ D and RITERIA FOR UNIVALENCE AND QUASICONFORMAL EXTENSION 9 z = z n + r n ζ ∈ ∆ n . Note that r n λ ∆ n ( z ) = λ D ( ζ ) and that λ ∆ n ( z ) ≥ λ Ω ( z ), for z ∈ ∆ n , since ∆ n ⊂ Ω. Let F n ( ζ ) = f ( z ) − f ( z n ) r n h ′ ( z n )and compute its dilatation ω n ( ζ ) = µ n ω ( z ), where µ n = h ′ ( z n ) /h ′ ( z n ) ∈ T .Compute also S F n ( ζ ) = r n S f ( z ). We have that | S F n ( ζ ) | λ D ( ζ ) = | S f ( z ) | λ ∆ n ( z ) ≤ | S f ( z ) | λ Ω ( z ) , so that k S F n k D ≤ k S f k Ω ≤ t and, therefore, we find that F n belongs to F t ( D ). Recalling the expression (11)we have that k ω ∗ n k ≤ R t ( D ). A calculation shows that ω ∗ n ( ζ ) = ω ′ n ( ζ ) λ D ( ζ )(1 − | ω n ( ζ ) | ) = µ n ω ′ ( z ) λ ∆ n ( z )(1 − | ω ( z ) | ) = µ n λ Ω ( z ) λ ∆ n ( z ) ω ∗ ( z )and, in particular, that | ω ∗ n (0) | = r n λ Ω ( z n ) | ω ∗ ( z n ) | . In view of Koebe’s 1/4-theorem (12) we have that r n λ Ω ( z n ) ≥ /
4. Therefore, | ω ∗ ( z n ) | ≤ | ω ∗ n (0) | ≤ k ω ∗ n k ≤ R t ( D ) . The proof is completed upon letting n → ∞ . (cid:3) Proof of Theorem 1
Without loss of generality we may assume that 0 ∈ Ω and that f ∈ F t (Ω) , t ≥ k Sh k Ω < ∞ . Moreover, since the classes F t (Ω)are nested and increasing with t we have that k Sh k Ω is uniformly bounded for f ∈ F t (Ω) and small t , that is, k Sh k Ω ≤ M for some constant M > t ≤
1. From the expression (3) we get | Sh | ≤ | S f | + | ω ′ | − | ω | (cid:12)(cid:12)(cid:12)(cid:12) h ′′ h ′ (cid:12)(cid:12)(cid:12)(cid:12) + | ω ′′ | − | ω | + 32 (cid:18) | ω ′ | − | ω | (cid:19) . Propositions 3 and 4, along with Koebe’s 1/4-theorem (12) yield | Sh ( z ) | λ Ω ( z ) ≤ k S f k Ω + 2 | ω ∗ ( z ) | d ( z ) λ Ω ( z ) r k Sh k Ω C k ω ∗ k d ( z ) λ Ω ( z ) + 32 | ω ∗ ( z ) | ≤ t + 8 k ω ∗ k r M C k ω ∗ k + 32 k ω ∗ k . Since k ω ∗ k ≤ k Sh k Ω ≤ t + ˆ CR t (Ω)for some constant ˆ C >
0. Hence, k Sh k Ω → t → + , in view of Lemma 5.Since σ (Ω) > t > k Sh k Ω < σ (Ω), so that h is univalent in Ω. Moreover, h has a quasiconformal extension to C byTheorem A, which we will denote by e h .Using the affine invariance of F t (Ω), the above calculation can be repeated forthe transform A a f , given in (9), for any a ∈ D . Thus, the analytic part of A a f and, therefore, h a = h + ag is univalent in Ω for every a ∈ D . Letting | a | → − and using Hurwitz’ theorem we get that h a is univalent in Ω for every a ∈ T ,since it can not be constant due to its normalization h ′ a (0) = 1 + ag ′ (0) = 0. Weshow that f is injective in Ω by contradiction. Let z , z ∈ Ω be distinct, andsuch that f ( z ) = f ( z ). Since h is injective, we have that h ( z ) = h ( z ). Setting θ = arg (cid:0) h ( z ) − h ( z ) (cid:1) we see that R ∋ e − iθ (cid:0) h ( z ) − h ( z ) (cid:1) = e − iθ (cid:0) g ( z ) − g ( z ) (cid:1) = e iθ (cid:0) g ( z ) − g ( z ) (cid:1) , (13)from which we get that h + e iθ g is not injective, a contradiction.Moreover, h a has a quasiconformal extension f h a to C for every a ∈ D . Inview of Theorem 5.3 in [15, Ch.II, § a ∈ T the limit function f h a iseither constant, or takes two values, or is quasiconformal. The first two cases arediscarded by the normalization at the origin and, therefore, f h a is quasiconformalin C for every a ∈ D . Now we define e f = e h + e g, where e h = f h and e g = f h − f h . It is clear that e f is a continuous (with respect tothe spherical metric) extension of f to C . It remains to show that e f is injectivein C , since the continuity of e f − may then be obtained by a general result, seeTheorem 5.6 in [17, § z , z ∈ C be distinct, and such that ℓ = e f ( z ) = e f ( z ) ∈ C . Since e h is injective, we have that e h ( z ) = e h ( z ). If both e h ( z ) and e h ( z ) are finite then we proceed as in (13), setting θ = arg (cid:0)e h ( z ) − e h ( z ) (cid:1) and seeing that e h + e iθ e g is not injective, a contradiction. If one of e h ( z ) and e h ( z ) is finite and the other is infinite, we may assume without loss of generalitythat e h ( z ) = ∞ and e h ( z ) = ∞ . Let γ θ be the pre-image of the ray { Re iθ : R ≥ } , θ ∈ [0 , π ), of the function e h . Clearly, γ θ is a simple curve with one endpointat the origin and the other at z . We write e h + e g = e f + e h − e h and see that for z ∈ γ θ we have that e h ( z ) − e h ( z ) = Re iθ (1 − e − iθ ). We distinguishtwo cases: ℓ (the common value of e f at z and z ) being finite or infinite. If ℓ = ∞ then lim γ θ ∋ z → z e h ( z ) + e g ( z ) = (cid:26) ℓ, if θ = 0 , π, ∞ , if θ = 0 , π, which is a contradiction since e h + e g is continuous in C . If ℓ = ∞ we take limit when z → z along γ (so that e h ( z ) = R ) in order to compute that ( e h + e g )( z ) = ∞ .But, on the other hand, we have that ( e h + e g )( z ) = ∞ , since e h ( z ) = ∞ and e g ( z ) = ∞ . This is a contradiction because e h + e g is injective. RITERIA FOR UNIVALENCE AND QUASICONFORMAL EXTENSION 11 Proof of Theorem 3
Let ϕ : D → Ω be a Riemann map of Ω with ϕ (0) = 0 and consider Ω r = ϕ ( {| z | < r } ) for r <
1. We set γ r = ∂ Ω r and note that, since Ω is a b K -quasidiskfor some b K ≥
1, it follows (trivially) that γ r is a b K -quasicircle for every r <
1. Wealso set Γ r = f ( γ r ) and claim that it is a K -quasicircle, with K ≥ r . Once we prove this claim we may then consider λ r and Λ r to be b K - and K -quasiconformal reflecions across γ r and Γ r , respectively, and setting e f r ( z ) = (cid:26) f ( z ) , if z ∈ Ω r , Λ r ◦ f ◦ λ r ( z ) , if z ∈ C \ Ω r , we see that this is a (cid:16) K d − d b K (cid:17) -quasiconformal mapping in the Riemann sphere.Letting r →
1, and in view of Theorem 5.3 in [15, Ch.II, § f to C .Let w j ∈ Γ r , j = 1 , , ,
4, be distinct points. Our claim that Γ r is a K -quasicircle, with K ≥ r , will be proved by showing that thecross-ratio of the points w j is bounded by a uniform constant, independent of r .Since f is injective in Ω by Theorem 1, there exist exactly four points z j ∈ γ r forwhich w j = f ( z j ). For convenience, we will write h j = h ( z j ) and g j = g ( z j ). Wehave w i − w j = h i − h j + g i − g j = ( h i − h j )(1 + µ ij A ij ) , where A ij = g i − g j h i − h j and µ ij = g i − g j g i − g j ∈ T (we may set µ ij = 1 if g i = g j ). We havethat | ( w , w , w , w ) | = | ( h , h , h , h ) | (cid:12)(cid:12)(cid:12)(cid:12) (1 + µ A )(1 + µ A )(1 + µ A )(1 + µ A ) (cid:12)(cid:12)(cid:12)(cid:12) . (14)Since h (Ω) is a quasidisk (in view of the proof of Theorem 1), we have that | ( h , h , h , h ) | ≤ M for some absolute constant M ≥ h a = h + ag is univalentin Ω for every a ∈ D . We will now expand the range of | a | for which this holds.Let δ ∈ (1 , /d ) and consider 1 < | a | ≤ δ . We compute h ′ a = h ′ (1 + aω ) and h ′′ a h ′ a = h ′′ h ′ + aω ′ aω , so that Sh a = Sh − aω ′ aω h ′′ h ′ + aω ′′ aω − (cid:18) aω ′ aω (cid:19) . By formula (3) and a straightforward computation we arrive at Sh a = S f + a + ω aω " ω ′′ − | ω | − ω ′ − | ω | h ′′ h ′ + 32 (cid:18) ω ′ − | ω | (cid:19) (cid:18) ω − a (1 − | ω | )1 + aω (cid:19) . In Ω, we have that (cid:12)(cid:12)(cid:12)(cid:12) a + ω aω (cid:12)(cid:12)(cid:12)(cid:12) ≤ max | ζ | = d (cid:12)(cid:12)(cid:12)(cid:12) a + ζ aζ (cid:12)(cid:12)(cid:12)(cid:12) = | a | − d − | a | d ≤ δ − d − δd = C, where the first of these inequalities follows from the maximum principle. More-over, we have that (cid:12)(cid:12)(cid:12)(cid:12) ω − a (1 − | ω | )1 + aω (cid:12)(cid:12)(cid:12)(cid:12) ≤ d + | a | (1 − | ω | )1 − | aω | ≤ d + | a | (1 − d )1 − | a | d ≤ d + δ (1 − d )1 − δd = C ′ . Hence, we get that | Sh a | ≤ | S f | + C " | ω ′′ | − | ω | + | ω ′ | − | ω | (cid:12)(cid:12)(cid:12)(cid:12) h ′′ h ′ (cid:12)(cid:12)(cid:12)(cid:12) + 3 C ′ (cid:18) | ω ′ | − | ω | (cid:19) . We assume that f ∈ F t (Ω), for t ≥
0, and, working as in the proof of Theorem 1,we use Propositions 3, 4 and Koebe’s 1/4-theorem (12) in order to obtain k Sh a k Ω ≤ t + ˆ CR t (Ω)for some constant ˆ C >
0. Choosing t > t + ˆ CR t (Ω) = σ (Ω) weconclude that h a is univalent.We fix α ∈ Ω and consider the generalized dilatation ψ α ( z ) = ( g ( z ) − g ( α ) h ( z ) − h ( α ) , if z ∈ Ω \{ α } ,ω ( α ) , if z = α. Since h is injective in Ω it is clear that ψ α is holomorphic in Ω. We claim that S = max z ∈ Ω r | ψ α ( z ) | ≤ δ . Note that | ω ( α ) | ≤ d < /δ so that if, otherwise, S > /δ , then there would exista point z ∈ Ω \{ α } for which ψ α ( z ) = e iθ /δ , for some θ ∈ R . This shows thatthe values of the function h − e − iθ δg at the points α and z would coincide, whichis a contradiction since this is a univalent function.Hence we may bound the terms in (14) as | A ij | ≤ /δ in order to obtain | ( w , w , w , w ) | ≤ M (1 + | A | )(1 + | A | )(1 − | A | )(1 − | A | ) ≤ M (cid:18) /δ − /δ (cid:19) , with which we finish the proof.6. Finitely connected domains
Let Ω be a domain in C . A collection D of domains D ⊂ Ω is called a quasicon-formal decomposition of Ω if each D is a quasidisk and any two points z , z ∈ Ωlie in the closure of some D ∈ D . This definition along with the following coveringlemma were given by Osgood in [18]. Lemma C ([18]) . If Ω is a finitely connected domain and each component of ∂ Ω is either a point or a quasicircle then Ω is quasiconformally decomposable. The proof in [18] provides an explicit finite decomposition.
RITERIA FOR UNIVALENCE AND QUASICONFORMAL EXTENSION 13
Proof of Therorem 2.
We prove the direction (i) ⇒ (iii). The domain Ω is quasi-conformally decomposable by a collection D in view of Lemma C. By Theorem 1,each of the quasidisks D in D has positive harmonic inner radius. Let0 < c ≤ min D ∈ D σ H ( D )and consider f to be a harmonic mapping in Ω that satisfies k S f k Ω ≤ c . If f ( z ) = f ( z ) for two distinct points z , z in Ω then z , z ∈ D , for some quasidisk D from the collection D . The domain monotonicity for the hyperbolic metricshows that λ D ( z ) ≥ λ Ω ( z ) for all z ∈ D and, therefore, that k S f k D ≤ k S f k Ω ≤ c. But now the homeomorphic extension of Theorem 1 shows that if c is sufficientlysmall then f is injective up to the boundary of D , a contradiction. (cid:3) We note that if we strengthen the definition of quasiconformal decompositionso that any two points z , z ∈ Ω lie in some quasidisk D (not its closure) froma collection D , then the construction in [18] can be modified so that Lemma Cstill holds. Had we followed this line of reasoning then we would not need thehomeomorphic extension of Theorem 1, but only the univalence criterion fromits statement. Acknowledgements . I would like to thank professor Martin Chuaqui for hisinsightful comments in our many discussions.
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