Affine and linear invariant families of harmonic mappings
aa r X i v : . [ m a t h . C V ] M a y AFFINE AND LINEAR INVARIANT FAMILIES OFHARMONIC MAPPINGS
MARTIN CHUAQUI, RODRIGO HERN ´ANDEZ, AND MAR´IA J. MART´IN
Abstract.
We study the order of affine and linear invariant familiesof planar harmonic mappings in the unit disk and determine the orderof the family of mappings with bounded Schwarzian norm. The resultshows that finding the order of the class S H of univalent harmonic map-pings can be formulated as a question about Schwarzian norm and, inparticular, our result shows consistency between the conjectured orderof S H and the Schwarzian norm of the harmonic Koebe function. Introduction
The purpose of this paper is to study certain affine and linear invariantfamilies of planar harmonic mappings defined in the unit disk D , with aspecial interest in the family of mappings with bounded Schwarzian norm.The fundamental aspects of linear invariant families of harmonic mappingswere studied in [20], while linear invariant families of holomorphic mappingswere introduced by Pommerenke in [17]. Several important properties ofsuch families of either holomorphic or harmonic mappings depend on itsorder, namely the optimal bound for the second Taylor coefficient of theholomorphic part of the mappings considered. The order of the class ofholomorphic mappings with bounded Schwarzian norm can be determinedby means of a variational method that gives a relation for the second andthird order coefficients of an extremal mapping [17]. Our motivation in thispaper stems from the still unresolved problem of determining the order of thefamily S H of normalized univalent harmonic mappings. In this direction, weare able to apply the variational approach that leads to the Marty relationsto determine the order of the family of harmonic mappings with a givenbound for the Schwarzian derivative. This seems relevant because the orderof the class S of normalized univalent holomorphic mappings can be derived Date : July 19, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Univalent function, harmonic mapping, Schwarzian derivative,affine and linear invariant family, order.The authors are partially supported by grants Fondecyt 1110160 and 1110321, Chile.The second author also thankfully acknowledges partial support from Faculty of Forestryand Sciences UEF, Finland (930349). The third author is supported by Academy ofFinland grant 268009 and by Spanish MINECO Research Project MTM2012-37436-C02-02. from the above coefficient relation and the well-known Schwarzian boundfor the class. Our result shows consistency between the conjectured valuesfor order in S H and in S H and the Schwarzian norm of the harmonic Koebefunction, a natural candidate for maximizing the Schwarzian norm in S H .The hyperbolic norm of the dilatations of the harmonic mappings enter inour analysis in an unexpected way, and turn out to be intimately related tothe order of the family. The construction of an extremal mapping for ourmain result is not elementary and depends on a subtle interplay between thesecond coefficient and the hyperbolic norm. In another direction, we alsoestablish the sharp bound for the Schwarzian norm of certain importantfamilies of harmonic mappings for which the order was already known.The Schwarzian derivative of a locally univalent analytic function on adomain in the complex plane is Sf = (cid:18) f ′′ f ′ (cid:19) ′ − (cid:18) f ′′ f ′ (cid:19) . The role of Sf in the study of univalence, distortion, and extensions of f has been developed extensively in the literature (see, e.g. , [2, 6, 12, 15]).Two of the main properties of the Schwarzian derivative are the following:i) Sf = 0 if and only if f is a M¨obius transformation.ii) Whenever the composition f ◦ g is well-defined, the chain rule holds: S ( f ◦ g ) = ( Sf ◦ g ) · ( g ′ ) + Sg .
In the case when f is locally univalent in the unit disk D , the Schwarziannorm k Sf k = sup | z | < | Sf ( z ) | · (1 − | z | ) turns out to be invariant under post-compositions with automorphisms σ ofthe disk. In other words, for any such functions f and σ , k S ( f ◦ σ ) k = k Sf k . In [17, 18] Pommerenke studied and carried through a detailed analysis ofthe so-called linear invariant families ; that is, families of locally univalentholomorphic functions f in the unit disk normalized by the conditions f (0) =1 − f ′ (0) = 0 and which are closed under the transformation F ζ ( z ) = f (cid:18) ζ + z ζz (cid:19) − f ( ζ )(1 − | ζ | ) f ′ ( ζ ) , ζ ∈ D . Several important properties, such as growth, covering, and distortion aredetermined by the order of a linear invariant family F defined by α ( F ) = sup f ∈F a ( f ) = 12 sup f ∈F | f ′′ (0) | . FFINE AND LINEAR INVARIANT FAMILIES 3
For example, the order of the important class S of normalized univalentmappings in D is 2. We refer the reader to the books [4] or [19] for moredetails related to the class S .In [17], Pommerenke proves the following theorem regarding the linearinvariant family H λ of normalized locally univalent analytic functions f inthe unit disk with k Sf k ≤ λ . Theorem A . The order of the family H λ is given by α ( H λ ) = r λ . It is a straightforward calculation to show that given λ ≥
0, the function(1) ϕ a ( z ) = 12 a (cid:20)(cid:18) z − z (cid:19) a − (cid:21) , | z | < , a = r λ , belongs to H λ and satisfies 12 | ϕ ′′ a (0) | = r λ . Whenever f ∈ S , its Schwarzian norm is bounded by 6. Therefore, S ⊂H so that by Theorem A we get | a | ≤ f ∈ S , although the boundwill hold also for the non-univalent mappings in H .A planar harmonic mapping in a domain Ω ⊂ C is a complex-valuedfunction w = f ( z ) = u ( z ) + iv ( z ), z = x + iy , which is harmonic, thatis, ∆ u = ∆ v = 0. When Ω is simply connected, the mapping f has acanonical decomposition f = h + g , where h and g are analytic in Ω. Asis usual, we call h the analytic part of f and g the co-analytic part of f .The harmonic mapping f is analytic if and only if g is constant. Lewy [13]proved that a harmonic mapping is locally univalent in a domain Ω if andonly if its Jacobian does not vanish. In terms of the canonical decomposition f = h + g , the Jacobian is given by | h ′ | − | g ′ | , and thus, a locally univalentharmonic mapping in a simply connected domain Ω will be sense-preserving or sense-reversing according to whether | h ′ | > | g ′ | or | g ′ | > | h ′ | in Ω. Notethat f = h + g is sense-preserving in Ω if and only if h ′ does not vanish andthe (second complex) dilatation ω = g ′ /h ′ has the property that | ω | < F be a family of sense-preserving harmonic mappings f = h + g in D , normalized with h (0) = g (0) = 0 and h ′ (0) = 1. The family is said to be affine and linear invariant ( AL family ) if it closed under the two operationsof Koebe transform and affine change :(2) K ζ ( f )( z ) = f (cid:18) z + ζ ζz (cid:19) − f ( ζ )(1 − | ζ | ) h ′ ( ζ ) , | ζ | < , M. CHUAQUI, R. HERN ´ANDEZ, AND M. J. MART´IN and(3) A ε ( f )( z ) = f ( z ) − εf ( z )1 − εg ′ (0) , | ε | < . Sheil-Small [20] offers an in depth study of affine and linear invariant families F of harmonic mappings in D . The order of the AL family, given by α ( F ) = sup f ∈F | a ( f ) | = 12 sup f ∈F | h ′′ (0) | , plays once more a special role in the analysis.A special example of affine and linear invariant family is the class S H of (normalized) sense-preserving harmonic mappings which are univalent inthe unit disk. As it is usual, we use S H to denote the family of functions f = h + g ∈ S H with g ′ (0) = 0. It is conjectured that the second Taylorcoefficient of the analytic part h of any function in S H is bounded by 5 / S H is equalto 3. The analytic part of the so-called harmonic Koebe function K ∈ S H (introduced by Clunie and Sheil-Small in [3]) has second coefficient equal to5 / S f of locally univalent harmonic mappings, which serves as a complement tothe definition found in [1]. The requirement of the latter that the dilatationbe a square has been replaced in the former by the local univalence. In bothcases a chain rule is in order, which shows that for mappings in D the norm || S f || defined as before is invariant under automorphisms of the disk. Butonly the Schwarzian S f introduced in [10] is invariant under affine changes af + bf , | a | 6 = | b | . As a result, the family F λ of sense-preserving harmonicmappings f = h + g in D , with h (0) = g (0) = 0 , h ′ (0) = 1 and || S f || ≤ λ , isaffine and linear invariant. We let F λ = { f ∈ F λ : g ′ (0) = 0 } .The hyperbolic norm of the dilatation ω of a sense-preserving harmonicmapping in D is given by k ω ∗ k = sup z ∈ D | ω ′ ( z ) | · (1 − | z | )1 − | ω ( z ) | , and will play a distinguished role in our analysis. Observe that k ω ∗ k ≤ λ ≥ / f ∈ F λ with a dilatation of hyperbolic norm equal to 1.Moreover, for λ < / F λ will be strictly less than 1.Let us denote by A λ (resp. A λ ) the set of admissible dilatations of func-tions f ∈ F λ (resp. F λ ); i.e., ω ∈ A λ (or A λ ) if there exists a harmonicmapping f = h + g ∈ F λ ( F λ ) with dilatation ω . The main purpose of thisarticle is to show the following generalization of Theorem A. FFINE AND LINEAR INVARIANT FAMILIES 5
Theorem . The order of F λ is given by α ( F λ ) = s λ f ∈F λ | g ′′ (0) | + 12 sup f ∈F λ | g ′′ (0) | (4) = s λ ω ∈A λ k ω ∗ k + 12 sup ω ∈A λ k ω ∗ k . (5) Furthermore,
12 sup f ∈F λ | h ′′ (0) | = s λ f ∈F λ | g ′′ (0) | (6) = s λ ω ∈A λ | ω ′ (0) | . (7)It was proved in [10] that the Schwarzian norm of the harmonic Koebefunction equals 19 /
2. For λ = 19 / f ∈F λ | h ′′ (0) | = 52 , which would show that the order of S H is equal to 3 provided the harmonicKoebe function was extremal in the class for the Schwarzian norm.1. Schwarzian derivative
Let f = h + g be a locally univalent harmonic mapping in a simplyconnected domain Ω with dilatation ω = g ′ /h ′ . In [10], the Schwarzianderivative S f of such a function f was defined. If f is sense-preserving, S f is given by(8) S f = Sh + ω − | ω | (cid:18) h ′′ h ′ ω ′ − ω ′′ (cid:19) − (cid:18) ω ′ ω − | ω | (cid:19) . Several properties of this operator are the following:(i) S f ≡ f = αT + βT , where | α | 6 = | β | and T is aM¨obius transformation of the form T ( z ) = az + bcz + d , ad − bc = 0 . (ii) Whenever f is a sense-preserving harmonic mapping and φ is ananalytic function such that the composition f ◦ φ is well-defined, theSchwarzian derivative of f ◦ φ can be computed using the chain rule S f ◦ φ = S f ( φ ) · ( φ ′ ) + Sφ . (iii) For any affine mapping L ( z ) = az + bz with | a | 6 = | b | , we have that S L ◦ f = S f . Note that L is sense-preserving if and only if | b | < | a | . M. CHUAQUI, R. HERN ´ANDEZ, AND M. J. MART´IN
Consider now a sense-preserving harmonic mapping f in the unit disk.Using the chain rule, we can see that for each z ∈ D | S f ( z ) | = | S ( f ◦ σ z ) (0) | · (1 − | z | ) , where σ z is any automorphism of the unit disk with σ z (0) = z . The Schwarzian norm k S f k of f is defined by k S f k = sup z ∈ D | S f ( z ) | · (1 − | z | ) . It is easy to check (using the chain rule again and the Schwarz-Picklemma) that k S f ◦ σ k = k S f k for any automorphism of the unit disk σ . Forfurther properties of S f and the motivation for this definition, see [10].2. Affine and Linear Invariant Families
Let S H denote the family of sense-preserving univalent harmonic map-pings f = h + g on D normalized by h (0) = 0, h ′ (0) = 1, and g (0) = 0.This family is affine and linear invariant. As usual, we use S H to denote thesubclass of functions in S H that satisfy the further normalization g ′ (0) = 0.The family S H is normal and S H is compact (see [3] or [5]). Analogousresults are obtained when dealing with the families C H and C H of convexharmonic mappings in S H and S H , respectively.Other examples of AL families of sense-preserving harmonic mappings arethe stable harmonic univalent ( SHU ) and the stable harmonic convex (
SHC )classes. A function f = h + g ∈ S H is SHU (resp.
SHC ) if h + λg is univalent(convex) for every | λ | = 1. These classes are linear invariant, and also affinebecause univalence or convexity are preserved under the affine changes A ε as in (3). An important observation is that if the harmonic mapping f has dilatation ω , then F = A ω (0) ( f ) will have a dilatation vanishing at theorigin.It is easy to check that α ( SHU )= 2 and α ( SHC )= 1 (see [8]). In the nexttheorem, we obtain sharp bounds for the Schwarzian norm of functions inthese classes.
Theorem . Let f = h + g be a locally univalent harmonic mapping definedin D . (i) If f is a SHU mapping, then k S f k ≤ . (ii) If f is a SHC mapping, then k S f k ≤ .Both constants are sharp.Proof. It was shown in [8] that if f = h + g ∈ SHU , then h + ag is univalentfor all | a | <
1; in particular, h itself is univalent. Assume that there exists f ∈ SHU with k S f k > ω be its dilatation. Then, there is apoint ζ ∈ D such that | S f ( ζ ) | · (1 − | ζ | ) >
6. Using the chain rule for theSchwarzian derivative and the affine invariance, we see that(9) | S A ωζ (0) ( K ζ ( f )) (0) | = | S f ( ζ ) | · (1 − | ζ | ) > , FFINE AND LINEAR INVARIANT FAMILIES 7 where K ζ is the transformation defined by (2), ω ζ is the dilatation of thefunction K ζ ( f ), and A ω ζ (0) is as in (3). Since the dilatation of K ζ ( f ) atthe origin is ω ζ (0), we have that the dilatation of A ω ζ (0) ( K ζ ( f )) fixes theorigin. Let H denote the (univalent) analytic part of A ω ζ (0) ( K ζ ( f )); keepingin mind the definition (8) for the Schwarzian derivative, we see from (9) that | S A ωζ (0) ( K ζ ( f )) (0) | = | SH (0) | > , which contradicts the univalence of H . This proves statement (i). The proofof (ii) follows the same argument, except for the fact that convex analyticmappings have Schwarzian norm bounded by 2 [16].To prove that both constants are sharp, it is enough to consider theanalytic functions k ( z ) = z (1 − z ) and s ( z ) = 12 log (cid:18) z − z (cid:19) that belong to the families of SHU and
SHC mappings and have Schwarziannorms k S k k = 6 and k S s k = 2, respectively. (cid:3) Admissible Dilatations
In this section, we review some of the properties of hyperbolic derivativesof self-maps of the unit disk and determine the relation between hyperbolicnorms of admissible dilatations in F λ and the parameter λ itself.3.1. The hyperbolic derivative.
Let ω be a self-map of the unit disk,this is, an analytic function in D with ω ( D ) ⊂ D . The hyperbolic derivative of such function ω is ω ∗ ( z ) = ω ′ ( z ) · (1 − | z | )1 − | ω ( z ) | , z ∈ D . From Schwarz’s lemma we see that | ω ∗ | ≤ D , and that if there exists z ∈ D with | ω ∗ ( z ) | = 1, then ω is an automorphism of the unit diskand | ω ∗ | ≡ D . Of course, there are self-maps of D with hyperbolicnorm equal to 1 which are not automorphisms. There are examples suchas ω ( z ) = ( z + 1) /
2, but also, every finite Blaschke product has hyperbolicnorm equal to 1 [7]. See also [14] for other examples.Given two self-maps ω and ϕ of the unit disk, the chain rule for thehyperbolic derivative holds:( ϕ ◦ ω ) ∗ ( z ) = ϕ ∗ ( ω ( z )) · ω ∗ ( z ) . In particular, if σ is an automorphism of D , then | ( σ ◦ ω ) ∗ | ≡ | ω ∗ | in theunit disk, hence k ( σ ◦ ω ) ∗ k = k ω k .Just like in [9], the so-called lens-maps ℓ α will be of particular interest.For 0 < α <
1, the mapping ℓ α is defined by(10) ℓ α ( z ) = ℓ ( z ) α − ℓ ( z ) α + 1 , M. CHUAQUI, R. HERN ´ANDEZ, AND M. J. MART´IN where ℓ ( z ) = (1 + z ) / (1 − z ). The hyperbolic norm of ℓ α was computed in[11] to be k ℓ ∗ α k = α . Moreover, | ℓ ∗ α ( r ) | = α for all real numbers 0 ≤ r < Admissible dilatations and norms.
Recall that we say that a self-map ω of the unit disk belongs to the family of admissible dilatations A λ if there exists f ∈ F λ with dilatation ω . For any such f and any α ∈ D ,the affine transformation f α = A − α ( f ) = f + αf ∈ F λ . Its dilatation ω α isgiven by ω α = σ α ◦ ω , where σ α is the automorphism in the unit disk defined by(11) σ α ( z ) = α + z αz , z ∈ D . In other words, whenever ω ∈ A λ and α ∈ D , then σ α ◦ ω ∈ A λ .On the other hand, if f ∈ F λ that has dilatation ω satisfying ω (0) = α ,then F = f − αf − | α | ∈ F λ , with a resulting new dilatation ω F = σ − α ◦ ω = σ − α ◦ ω . These relationsestablish a correspondence between the families A λ and A λ .It is easy to verify that for any value of λ , there exists ω ∈ A λ with k ω ∗ k = 0 (just consider the identity function I ( z ) = z in the unit disk whichbelongs to F λ for all λ ≥ λ for which a dilatation in A λ can have hyperbolic norm 1. Theorem . The following conditions are equivalent. (i) λ ≥ / . (ii) There exists ω ∈ A λ with | ω ′ (0) | = 1 . (iii) The set { λ · I : | λ | = 1 } is contained in A λ . In particular, the identityfunction I is an admissible dilatation in F λ . (iv) Every automorphism σ of the unit disk is an admissible dilatation in F λ . (v) There exist ω ∈ A λ with k ω ∗ k = 1 .Proof. The scheme of the proof is to show that (i) ⇒ (ii) ⇒ (iii) ⇒ (i).Then we prove that (iii) ⇐⇒ (iv), and finally we see that (iv) ⇒ (v) ⇒ (ii).To show that (i) ⇒ (ii), we consider the function f = z + z . Note thatthe dilatation ω of f equals the identity function I so that ω ′ (0) = 1. Since S f ( z ) = − z − | z | ) , we have k S f k = 3 /
2. Thus, f ∈ F λ for all λ ≥ / ⇒ (iii). To do so, just note that given anyharmonic function f = h + g ∈ F λ with dilatation ω and any µ ∈ ∂ D , thefunctions f µ = h + µg belong to F λ as well since S f = S f µ for all such µ andthe dilatation ω µ of f µ is equal to ω µ ( z ) = µ ω ( z ) (which satisfies ω µ (0) = 0). FFINE AND LINEAR INVARIANT FAMILIES 9
Now, if we assume that there exists ω ∈ A λ with | ω ′ (0) | = 1 we have, bythe Schwarz lemma, that ω is a rotation of the disk (i.e. ω ( z ) = λz for some | λ | = 1). Therefore, using those functions f µ we immediately get that (iii)holds.Suppose that (iii) is satisfied. This is, there is a function f = h + g ∈ F λ with dilatation ω = I . We are going to prove that λ ≥ /
2. Indeed, we willobtain a stronger result in view of inequality (16) below. More specifically,we will see that (iii) implies that the order α ( F λ ) of the family F λ is 2 atleast, so that by (16) we obtain (i) (note that the supremum that appearsthere equals 1 whenever I ∈ A λ ) .Using the affine invariance property of F λ , we have that whenever f ∈F λ , the function F ε = A ε ( f ) = f − εf ∈ F λ for all ε ∈ D , where A ε is the transformation defined in (3). The analytic part in the canonicaldecomposition of F ε equals h ε = h − εg so that h ′ ε ( z ) = h ′ ( z ) · (1 − εI ).Therefore, just applying the same arguments that appear in the proof of thetheorem on [5, p. 97], we get that for all z ∈ D (12) | h ′ ( z ) | · | − εz | ≥ (1 − | z | ) α ( F λ ) − (1 + | z | ) α ( F λ )+1 whenever ε < | ε | ≤
1. Thus given any z = 0 in the unitdisk, we can choose ε = z/ | z | to get from (12) that | h ′ ( z ) | ≥ (1 − | z | ) α ( F λ ) − (1 + | z | ) α ( F λ )+1 , an inequality that obviously works for z = 0. As a consequence, since h islocally univalent in D , we see that the analytic function 1 /h ′ in the unit disksatisfies 1 | h ′ ( z ) | ≤ (1 + | z | ) α ( F λ )+1 (1 − | z | ) α ( F λ ) − , which implies, by the maximum modulus principle, that α ( F λ ) ≥ / | h ′ (0) | <
1, which is absurd since h ′ (0) = 1). Thisshows that (iii) ⇒ (i).We continue with the proof of the theorem by showing that (iii) ⇔ (iv).Note that since every rotation is an automorphism of the unit disk thatfixes the origin, only the implication (iii) ⇒ (iv) is needed to check theequivalence between these two statements. Let us assume then that the set { λI : | λ | = 1 } ⊂ A λ . Then, we can use that F λ is affine invariant and thefunctions f µ defined above to see that for any α in the unit disk and any λ, µ ∈ ∂ D , the functions µσ α ◦ ( λI ) = µ · α + λz αλz = µλ · λα + z λαz ∈ A λ . In other words, for any | η | = 1 and any β ∈ D , the mappings(13) η · β + z βz ∈ A λ . The Schwarz lemma says that any automorphism of the unit disk has theform (13) so that (iv) holds.Finally, we are show the equivalence between statements (iv) and (v).Once more, only one of the implications in the equivalence is non-trivial(recall that any automorphism of the unit disk has hyperbolic norm equalto one). Concretely, we just need to see that (v) ⇒ (iv). Since at this pointwe have proved that (i), (ii), (iii), and (iv) are equivalent, it suffices to checkthat (v) ⇒ (ii). To do so, let ω ∈ A λ have the property that k ω ∗ k = 1.Then, by the definition of the hyperbolic norm, there exists a sequence ofpoints in the unit disk { z n } , say, such thatlim n →∞ | ω ∗ ( z n ) | = 1 . The fact that ω ∈ A λ means that there is a function f = h + g ∈ F λ withdilatation ω . Using the transformations K z n and A ω n (0) defined by (2) and(3), respectively, where ω n is the dilatation of K z n ( f ), we obtain a sequenceof harmonic mappings f n = A ω n (0) ( K z n ( f ))with dilatations γ n = σ − ω n (0) ◦ ω n = σ − ω n (0) ◦ [ λ n · ( ω ◦ σ z n )] , where λ n = h ′ ( z n ) /h ′ ( z n ) and σ α is again the automorphism of D definedby (11).Note that γ n (0) = 0, so that each element in the sequence f n ∈ F λ .Moreover, a straightforward computation shows that we also have | γ ′ n (0) | = | ω ∗ ( z n ) | .Now, we argue as on [5, pp. 81-82] but in this case, instead of the argu-ment principle, we use the fact that whenever f n → f uniformly on compactsubsets of D we have that for each z in the unit disk S f n ( z ) → S f ( z ) . This shows that F λ is a normal and compact family. Thus, there exists asubsequence (that we rename { f n } again) of the sequence { f n } that con-verges to f ∈ F λ uniformly on compact subsets in the unit disk. Thedilatation γ of the limit function f is γ ( z ) = lim n →∞ γ n ( z ) , so that it satisfies | γ ′ (0) | = (cid:12)(cid:12)(cid:12) lim n →∞ γ ′ n (0) (cid:12)(cid:12)(cid:12) = lim n →∞ | ω ∗ ( z n ) | = 1 . This proves that (v) ⇒ (ii) and finishes the proof of the theorem. (cid:3) From Theorems 1 and 3 we derive the following important corollary.
FFINE AND LINEAR INVARIANT FAMILIES 11
Corollary . If λ ≥ / then α ( F λ ) = r λ and
12 sup f ∈F λ | h ′′ (0) | = r λ . Even though for λ < / ω ∈A λ k ω ∗ k , one can rewrite Theorem 1 in the form(14) sup ω ∈A λ k ω ∗ k = − α ( F λ ) + p α ( F λ ) − λ − . Proof of the Main Theorem
We will divide the proof of Theorem 1 in three different parts.4.1.
A lemma.
We begin by proving that (5) and (7) hold.
Lemma . For any positive real number λ , (i) sup f = h + g ∈F λ | g ′′ (0) | = sup ω ∈A λ | ω ′ (0) | . (ii) sup ω ∈A λ | ω ′ (0) | = sup ω ∈A λ k ω ∗ k = sup ω ∈A λ k ω ∗ k . Proof.
The proof of (i) is trivial: just recall that the dilatation ω of f = h + g ∈ F λ equals ω = g ′ /h ′ and satisfies ω (0) = 0. Thus, g ′ = ωh ′ andhence g ′′ (0) = ω ′ (0) h ′ (0) + ω (0) h ′′ (0) = ω ′ (0), since h ′ (0) = 1.To prove (ii), take an arbitrary f = h + g ∈ F λ with dilatation ω . Thetransformations (2) and (3) produce new functions that also belong to thisfamily. Concretely, given any such function f , we define the mappings f ζ = A ω ζ (0) ( K ζ ( f )) = h ζ + g ζ , ζ ∈ D , where ω ζ is the dilatation of K ζ ( f ). These mappings f ζ belong to F λ andhave dilatations γ ζ = σ − ω ζ (0) ◦ ω ζ ∈ A λ , where ω ζ = λ ζ · ( ω ◦ σ ζ ) for certain | λ ζ | = 1 and appropriate automorphisms σ α of the form (11). Since | γ ′ ζ (0) | = | ω ∗ ( ζ ) | and ζ is any arbitrary point in D , we conclude that the first equality in(ii) holds. The second inequality is an easy consequence of the fact thatthe correspondence between the families A λ and A λ is realized using pre-composition with automorphisms of the unit disk (an operation that pre-serves the hyperbolic norm). (cid:3) Upper bounds.
The aim of this section is to prove the following in-equalities:(15) 12 sup f ∈F λ | h ′′ (0) | ≤ s λ ω ∈A λ | ω ′ (0) | and(16) α ( F λ ) ≤ s λ ω ∈A λ k ω ∗ k + 12 sup ω ∈A λ k ω ∗ k . To prove (15), let us agree with the notation S = 12 sup f ∈F λ | h ′′ (0) | . Given f ∈ F λ , the function F defined by F ( z ) = λf ( λz ) ∈ F λ as well.Therefore, we see that S = 12 sup f ∈F λ Re { h ′′ (0) } . As it was mentioned in the proof of Theorem 3, the family F λ is normaland compact. Therefore, there exists a function f = h + g in F λ withdilatation ω , where h ( z ) = z + a z + a z + . . . , g ( z ) = b z + b z + . . . , such that Re { a } = | a | = S .
Take an arbitrary point ζ ∈ D and consider, once more, the transforma-tions (2) and (3) to produce the family F ζ = A ω ζ (0) ( K ζ ( f )) = h ∗ ζ + g ∗ ζ of functions that are in F λ .As it is shown on [5, p.102], the Taylor coefficients a ∗ n of h ∗ ζ satisfy a ∗ n = a n + [( n + 1) a n +1 − a a n ] ζ − [2 b b n + ( n − a n − ] ζ + o ( | ζ | ) . Then, we get that(17) 3 a − a − | b | − . Now, using the equations S f (0) = Sh (0) = 6( a − a ), we have by(17) that Re { S f (0) } = 4 | b | + 2 − { a } . Hence, bearing in mind S =Re { a } = | a | , we obtain S = | a | ≤ | S f (0) | + 1 + 2 | b | ≤ λ ω ∈A λ | ω ′ (0) | . FFINE AND LINEAR INVARIANT FAMILIES 13
This proves (15). To show that (16) holds we can argue as follows. Takeany function f = h + g ∈ F λ with dilatation ω and consider the affinetransformation A ω (0) as in (3) to get the function F = A ω (0) ( f ) = H + G ∈ F λ . Note that f = F + ω (0) F . Thus, the analytic part of f equals H + ω (0) G .Using also that ω ( D ) ⊂ D , and Lemma 1, we get | h ′′ (0) | = | H ′′ (0) + ω (0) G ′′ (0) | ≤ S + | ω (0) | sup f ∈F λ | g ′′ (0) |≤ S + sup ω ∈A λ k ω ∗ k , which gives (16).4.3. Equalities.
In order finish the proof of Theorem 1, we will show that(15) and (16) are actual equalities.To see that equality holds in (15), we will exhibit a function f = h + g ∈F λ with dilatation ω satisfying the following properties: ω ′ (0) = sup ω ∈A λ k ω ∗ k and h ′′ (0) = 2 · r λ ω ′ (0) . This will show that (6) holds. Furthermore, for any 0 < r <
1, the mappings f r = f + rf ∈ F λ and have analytic parts h r = h + rg , for which h ′′ r (0) = h ′′ (0) + rg ′′ (0) = sup f ∈F λ | h ′′ (0) | + rω ′ (0)= sup f ∈F λ | h ′′ (0) | + r sup ω ∈A λ k ω ∗ k→ sup f ∈F λ | h ′′ (0) | + sup ω ∈A λ k ω ∗ k , as r → − . This shows that equality also holds in (16), proving thus (4).The function f will be constructed in the next, final section.5. The mapping f Given λ ≥
0, let us denote by R λ = sup ω ∈A λ k ω ∗ k . If λ = 0, then by [10,Cor. 2] every function in F λ has the form h + ah , where a ∈ D and h ( z ) = z bz , | b | < . Hence R λ = 0. We now analyze the cases λ > Harmonic mappings in F λ when λ > . By Theorem 3, we havethat λ ≥ / R λ = 1. Equation (14) gives the exact value of R λ in terms of λ and α ( F λ ). The following example allows us to estimatethe value of R λ in terms of λ for 0 < λ < / < λ < /
2, we can write λ = 3 s / < s <
1. Considerthe functions f r = z + rz , where r ∈ (0 , f r equals(18) k S f r k = 32 sup z ∈ D r | z | (1 − | rz | ) · (1 − | z | ) ≤ r . Hence, f r ∈ F λ whenever r ≤ s . This implies that R λ ≥ r = || ω ∗ r || , where ω r is the dilatation of f r . In particular, this implies that if λ ∈ (0 , / R λ > r λ . Note that we have strict inequality in (19) since the supremum that appearsin (18) is strictly less than r . Also, that R λ > λ > The extremal.
Let us introduce the notation α = α ( F λ ) and R = R λ = sup {k ω ∗ k : ω ∈ A λ } . For λ > f = h + g ,where h and g solve the linear system of equations(20) (cid:26) h − g = ϕ a ω = g ′ /h ′ = ℓ R , h (0) = g (0) = 0 , where a = r λ R − R , and the function ϕ a is the generalized Koebe function defined by (1). Also, ℓ is the identity mapping I and for 0 < R < ℓ R is the lens-map (10). Notethat h ′ (0) = 1 − g ′ (0) = 0 and that f is a locally univalent mapping in theunit disk since the dilatation is a self-map of D and h is locally univalent.By (20), we have h ′ (1 − ℓ R ) = ϕ ′ a which implies h ′′ h ′ = ϕ ′′ a ϕ ′ a + ℓ ′ R − ℓ R , so that Sh = Sϕ a + ℓ ′′ R − ℓ R + 12 ( ℓ ′ R ) (1 − ℓ R ) − ℓ ′ R − ℓ R · ϕ ′′ a ϕ ′ a , which gives Sh (0) = 2(1 − a ) + R − aR = 2 + R − aR − a . Define the function ψ ( x ) = 2 + R − xR − x . FFINE AND LINEAR INVARIANT FAMILIES 15
Note that ψ r R − R ! = 0 . Also, that ψ ′ < a > r R − R , that ψ ( a ) <
0. This means that | Sh (0) | = 2 a + 2 aR − − R /
2, which iseasily seen to be equal to λ . Since ω (0) = 0, we also have that | Sf (0) | = λ .To show that f ∈ F λ , we just need to check λ = | S f (0) | = k S f k . To do so,we compute the Schwarzian derivative of f which, according to (8), equals S f = Sh + ℓ R − | ℓ R | (cid:18) h ′′ h ′ ℓ ′ R − ℓ ′′ R (cid:19) − (cid:18) ℓ R ℓ ′ R − | ℓ R | (cid:19) = Sϕ a + ℓ ′′ R − ℓ R + 12 ( ℓ ′ R ) (1 − ℓ R ) − ℓ ′ R − ℓ R · ϕ ′′ a ϕ ′ a (21) + ℓ R − | ℓ R | (cid:18) ϕ ′′ a ϕ ′ a ℓ ′ R + ( ℓ ′ R ) − ℓ R − ℓ ′′ R (cid:19) − (cid:18) ℓ R ℓ ′ R − | ℓ R | (cid:19) . Since for any complex number z ,11 − z − z − | z | = 1 − z (1 − z )(1 − | z | ) , we get from (21) S f = Sϕ a + 1 − ℓ R − ℓ R · ℓ ′′ R − | ℓ R | − − ℓ R − ℓ R · ℓ ′ R − | ℓ R | · ϕ ′′ a ϕ ′ a + ( ℓ ′ R ) · "(cid:18) ℓ R − | ℓ R | (1 − ℓ R )(1 − | ℓ R | ) (cid:19) − · (cid:18) ℓ R − | ℓ R | (cid:19) = Sϕ a + F R · (cid:18) ℓ ′′ R ℓ ′ R − ϕ ′′ a ϕ ′ a (cid:19) + F R · ℓ R − | ℓ R | − ℓ R , (22)where F R = 1 − ℓ R − ℓ R · ℓ ′ R − | ℓ R | . Let us write(23) w = (cid:18) z − z (cid:19) R and β = wRew . Then, F R ( z ) = Rβ/ (1 − z ), ℓ ′′ R ( z ) ℓ ′ R ( z ) − ϕ ′′ a ( z ) ϕ ′ a ( z ) = − a − z + 2 R − z · − w w , and 1 + 3 ℓ R − | ℓ R | − ℓ R = 5Re { w } − i Im { w } )1 + w = 8Re { w } w − . Thus, we get from (22) S f ( z ) · (1 − z ) = 2(1 − a ) − αRβ − R β
2+ 2 R β · − w w + 4 R β · Re { w } w = 2(1 − a ) + 2 R ( R − a ) β − R β , a formula that we use to show | S f ( z ) | · (1 − | z | ) = (cid:12)(cid:12)(cid:12)(cid:12) − a ) + 2 R ( R − a ) β − R β (cid:12)(cid:12)(cid:12)(cid:12) · (cid:18) − | z | | − z | (cid:19) . (24)Consider a real number γ with 0 ≤ γ < π/ C γ = (cid:26) z ∈ D : Arg (cid:18) z − z (cid:19) = γ (cid:27) . Note that C is equal to the real diameter ( − ,
1) and for γ = 0, C γ is acircular arc passing trough the points − Lemma . The quantity | S f ( z ) | · (1 − | z | ) is constant on the curves C γ , ≤ γ < π/ .Proof. Take any γ ∈ [0 , π/
2) and let z ∈ C γ . Then there exists a (positive)real number t such that 1 + z − z = te iγ . A straightforward computation gives β = t R e iRγ t R cos( Rγ ) = 1 + i tan( Rγ ) and 1 − | z | | − z | = cos γ . The proof of the lemma follows from (24). (cid:3)
Lemma 2 shows, in particular, that whenever r ∈ (0 , | S f ( r ) | (1 − r ) = | S f (0) | ≡ λ . Note that S f ( z ) = S f ( z ) for all z ∈ D . Moreover, it is easy to checkthat any radius { re iθ , < r < } , with 0 < θ < π , intersects every C γ with γ > k S f k = sup ≤ r< | S f ( ir ) | · (1 − r ) . Lemma . The Schwarzian norm of f equals k S f k = | S f (0) | = λ . FFINE AND LINEAR INVARIANT FAMILIES 17
Proof.
According to (25), it suffices to showsup ≤ r< | S f ( ir ) | · (1 − r ) = | S f (0) | . We use (24) to writeΦ( r ) = | S f ( ir ) | · (1 − | r | ) = (cid:12)(cid:12)(cid:12)(cid:12) − a ) + 2 R ( R − a ) β r − R β r (cid:12)(cid:12)(cid:12)(cid:12) · (cid:18) − r r (cid:19) (26) = | φ ◦ β r | · (cid:18) − r r (cid:19) , where φ ( x ) = A + Bx + Cx , with A = 2(1 − a ), B = 2 R ( R − a ), and C = − R /
2; and, by (23),(27) β r = 1 + i tan( Rγ r ) with cos γ r = (1 − r ) / (1 + r ) . We are to check(28) sup { Φ( r ) : 0 ≤ r < } = Φ(0) . Instead of proving (28), we consider the equivalent problem of showing(29) sup { Φ ( r ) : 0 ≤ r < } = Φ (0) . The advantage of this new reformulation is that, as the reader may check,we can write | φ ◦ β r | = e A + e B | β r | + e C | β r | , with e A = A + 2 AB + 4 AC , e B = B + 2 BC − AC , and e C = C . Note that e A + e B + e C = λ . In fact,(30) e A = 4(1 − a ) (cid:0) − ( a + R ) (cid:1) (with a + R >
1, by the definition of a ),(31) e B = 2 R (3 − aR − a − R ) , and e C = 9 R . Notice that by (27), the mapping r → γ r is increasing for r ∈ (0 , | β r | = 1 / cos( Rγ r ).Let us rename Ψ( r ) = Φ ( r ). To prove (29), we distinguish among thefollowing three cases. (i) λ = / . In this case, by Theorem 3, R = 1. We also have a = 1,which gives A = B = 0, C = − / e A = e B = 0, e C = 9 / r ) = (cid:16) e A + e B | β r | + e C | β r | (cid:17) · cos γ r = 94 · γ r · cos γ r = 94 = λ . This proves that (29) holds for λ = 3 / (ii) λ > / . By Theorem 3, we have R = 1. A straightforward calcula-tion shows that a >
1. Hence, using (30) and that a + R >
1, we obtain e A >
0. Now, since | β r | = 1 / cos γ r , we can writeΨ( r ) = (cid:16) e A + e B | β | + e C | β | (cid:17) · cos γ r = e A | β | + e B | β | + e C ! = e A cos ( γ r ) + e B cos ( γ r ) + e C .
Consider the function ψ ( x ) = e Ax + e Bx + e C, x ∈ (0 , r ) = ψ (cos ( γ r )) and that Ψ( r ) ≤ Ψ(0) if and only if ψ ( x ) ≤ ψ (1). To show that ψ ( x ) ≤ ψ (1) we argue as follows: the graph of ψ is a convex parabola and ψ (1) = λ > / ψ (0) since λ > /
2. Hence, the unique critical point x of ψ is a minimum and satisfies x <
1. We have the following possibilities.(i) The critical point x ≤
0, which implies that for all r ∈ [0 , ψ ( r ) ≤ ψ (1) = λ .(ii) The critical point x which is a minimum of ψ belongs to (0 , ≤ x ≤ ψ ( x ) = max { ψ (0) , ψ (1) } = max (cid:26) , λ (cid:27) = λ = ψ (1) , which shows that (29) holds also for λ > / (iii) λ < / . This is the most complicated case to analyze by far due tothe amount of parameters that we are to control. Note that we have that
R < a < e A <
0, while e B, e C >
0. Recall that β r = 1 + i tan( Rγ r ) andthat cos γ r = (1 − r ) / (1 + r ), which gives that | β r | = 1 / cos( Rγ r ) and thatthe correspondence r → γ r is an (strictly) increasing function in r ∈ (0 , γ ′ r = ∂γ r /∂r > r ).We are to show that sup ≤ r< Ψ( r ) = Ψ(0). To do so, we will check thatthe derivative of Ψ is non-positive for all such r . For the convenience of thereader, we proceed in different steps. Step 1: There exists r ∈ (0 , such that Ψ ′ ( r ) < for all r ∈ (0 , r ) . As it was mentioned before, we can writeΨ( r ) = (cid:16) e A + e B | β r | + e C | β r | (cid:17) · cos γ r . Using this expression for Ψ, we compute its derivative to obtainΨ ′ ( r ) = (cid:16) e B | β r | + 4 e C | β r | (cid:17) · sin( Rγ r )cos ( Rγ r ) · Rγ ′ r · cos γ r − (cid:16) e A + e B | β r | + e C | β r | (cid:17) · γ r · sin γ r · γ ′ r , FFINE AND LINEAR INVARIANT FAMILIES 19 which can be written asΨ ′ ( r ) = − γ ′ r · cos γ r · tan γ r cos( Rγ r )(32) × (cid:16) e A cos( Rγ r ) + 2 e B | β r | + 2 e Bϕ r | β r | + 4 e Cϕ r | β r | (cid:17) , where ϕ r = 1 − R tan( Rγ r )tan γ r . Let us see that ϕ r is an (strictly) increasing function of r ∈ (0 , ϕ ′ r = − Rγ ′ r tan γ r · cos γ r · cos( Rγ r ) · (cid:18) R sin γ r cos( Rγ r ) − sin( Rγ r )cos γ r (cid:19) . Now, ϕ r is increasing if and only if(33) R sin γ r cos( Rγ r ) < sin( Rγ r )cos γ r ⇔ R sin(2 γ r ) − sin(2 Rγ r ) < . Define µ ( r ) = R sin(2 γ r ) − sin(2 Rγ r ) , r ∈ (0 , µ (0) = 0 and µ ′ ( r ) = 2 R (cos(2 γ r ) − cos(2 Rγ r )) · γ ′ r < < R <
1, which gives (33) and shows that ϕ r is increasing for all r ∈ (0 , Rγ r ) is decreasing in r so that 1 / cos( Rγ r ) = | β r | is increasing as well. Hence, the expression in the parentheses in (32) is anincreasing function of r ∈ (0 ,
1) (recall that e A < e B, e C > γ r · tan γ r increases with r for all 0 < r < r , wheresin( γ r ) = 1 /
2. This shows that as long as 0 < r < r , the function e ϕ ( r ) = − cos γ r · tan γ r cos( Rγ r ) · (cid:16) e A cos( Rγ r ) + 2 e B | β r | + 2 e Bϕ r | β r | + 4 e Cϕ r | β r | (cid:17) is decreasing. Since e ϕ (0) = 0, we get that e ϕ ( r ) < r ∈ (0 , r ). Inother words, keeping in mind that γ ′ r > r ∈ (0 , ′ ( r ) < r ∈ (0 , r ). (This implies, in particular, that r = 0is a local maximum of Ψ.) Step 2: There exists r ∈ (0 , such that Ψ ′ ( r ) < for all r ∈ ( r , . Recall that λ = 2 a + 2 aR − − R / . Using (26) and (27), we getΨ( r ) = (cid:12)(cid:12)(cid:12)(cid:12) − a ) + 2 R ( R − a ) β r − R β r (cid:12)(cid:12)(cid:12)(cid:12) · (cid:18) − r r (cid:19) = (cid:18) Re (cid:26) − a ) + 2 R ( R − a ) β r − R β r (cid:27)(cid:19) · cos γ r + (cid:18) Im (cid:26) − a ) + 2 R ( R − a ) β r − R β r (cid:27)(cid:19) · cos γ r = (cid:18) − λ + 3 R · tan ( Rγ r ) (cid:19) · cos γ r + (cid:0) R + 2 aR (cid:1) · tan ( Rγ r ) · cos γ r (34) = λ · cos γ r + K · tan ( Rγ r ) · cos γ r + 9 R · tan ( Rγ r ) · cos γ r , where K = R + 4 a R + 4 aR − R λ = 5 R R − a R − aR ≥ R R > , since a and R are less than one. Using (34) we see that the derivative of Ψwith respect to r equalsΨ ′ ( r ) = − λ · cos γ r · sin γ r · γ ′ r + K · R tan( Rγ r )cos ( Rγ r ) · cos γ r · γ ′ r − K · cos γ r · sin γ r · tan ( Rγ r ) · γ ′ r + 9 R · tan ( Rγ r )cos ( Rγ r ) · cos γ r · γ ′ r − R tan ( Rγ r ) · cos γ r · sin γ r · γ ′ r . Note that Ψ ′ ( r ) < K · R tan( Rγ r )cos ( Rγ r ) · cos γ r + 9 R · tan ( Rγ r )cos ( Rγ r ) · cos γ r < λ · sin γ r + 4 K · sin γ r · tan ( Rγ r ) + 9 R tan ( Rγ r ) · sin γ r , (35)an inequality the holds for all r ≥ r , say, since as r → γ r → π/ ′ ( r ) < r ∈ ( r , Step 3: Ψ has at most one critical point in (0 , . Note that oncewe check that the number of solutions of the equation Ψ ′ ( r ) = 0, r ∈ (0 , ′ ( r ) ≤ r ∈ (0 ,
1) (and hence Ψ is non-increasing in that interval).Observe that Ψ ′ = 0 if and only if the function in the parentheses in (32) is FFINE AND LINEAR INVARIANT FAMILIES 21 equal to zero. As was justified in Step 1, this function is increasing, whichproves our claim. (cid:3)
We summarize the previous analysis in the following proposition.
Proposition . If λ > then the function f ∈ F λ . This shows that (15) and (16) are equalities, and finishes the proof ofTheorem 1.
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E-mail address : [email protected] Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´a˜nez, Av. PadreHurtado 750, Vi˜na del Mar, Chile.
E-mail address : [email protected] Department of Physics and Mathematics, University of Eastern Finland,P.O. Box 111, FI-80101 Joensuu, Finland.
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