Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm
aa r X i v : . [ m a t h . C V ] J a n UNIFORMLY LOCALLY UNIVALENT HARMONIC MAPPINGSASSOCIATED WITH THE PRE-SCHWARZIAN NORM
GANG LIU AND SAMINATHAN PONNUSAMY
Abstract.
In this paper, we consider the class of uniformly locally univalent harmonicmappings in the unit disk and build a relationship between its pre-Schwarzian normand uniformly hyperbolic radius. Also, we establish eight ways of characterizing uni-formly locally univalent sense-preserving harmonic mappings. We also present somesharp distortions and growth estimates and investigate their connections with Hardyspaces. Finally, we study subordination principles of norm estimates. Introduction
Harmonic mappings play an important role in various branches of applied mathematicsincluding the study of liquid crystals, both in theory and in practice. There are manyclassical approaches to deal with harmonic maps in various settings. For example, A.Aleman and A. Constantin [3] developed tools using complex analytic theory and theunivalence of the labelling map to solve fluid flow problems in a surprisingly simple form.More recently, O. Constantin and M. J. Mart´ın [15] proposed a new approach to obtain acomplete solution to the problem of classifying all two dimensional ideal fluid flows withharmonic labelling maps. This approach is based on ideas from the theory of harmonicmappings by finding two harmonic maps with same Jacobians and illustrates the deeplinks between the fields of complex analysis and fluid mechanics. Investigations of thistype have prompted renewed interest in the study of sense-preserving harmonic mappings.The present article is concerned with Schwarzian and pre-Schwarzian norms defined inthe unit disk, and in particular, with certain important function spaces. In addition, weintroduce several new ideas and tools for a number of problems in the case of harmonicmappings.1.1.
Basic notations.
A complex-valued function f in the unit disk D = { z : | z | < } iscalled a harmonic mapping if it satisfies the Laplace equation ∆ f = 4 f zz = f xx + f yy = 0.It is known that f has a canonical representation f = h + g with g (0) = 0, where h and g are analytic functions in D and J f = | h ′ | − | g ′ | denotes the Jacobian of f . As is usual,we call h the analytic part of f and g the co-analytic part of f . Lewy [27] proved that f = h + g is locally univalent in D if and only if J f ( z ) = 0 in D . Without loss of generality,we consider harmonic mappings f that are sense-preserving, i.e. J f > | h ′ | > | g ′ | in D . In this case, its dilatation ω f = g ′ /h ′ has the property that | ω f | < Mathematics Subject Classification.
Primary: 30H10, 30H30, 30H35, 31A05; Secondary 30C55.
Key words and phrases.
Uniformly locally univalent, analytic functions, harmonic and stable harmonicmappings, pre-Schwarzian and Schwarzian, uniformly hyperbolic radius, distortion, Hardy space, Blochspace, BMOA, BMOH, subordination.File: LiuSamy3˙2016˙˙final.tex, printed: 12-10-2018, 4.09. D . Especially, if | ω f | ≤ k < D , then f is called a K − quasiconformal mapping, where K = (1 + k ) / (1 − k ). More details about planar harmonic mappings, may be found in themonograph of Duren [18] and in the survey article of Ponnusamy and Rasila [33].For the convenience of the reader, we first list down the following notations and termi-nologies whose precise definitions will be presented at appropriate places. • ULU (ULC) - uniformly locally univalent (uniformly locally convex) • SAULU - stable analytic uniformly locally univalent • SAULC - stable analytic uniformly locally convex • SHULU - stable harmonic uniformly locally univalent • SHULC - stable harmonic uniformly locally convex • SHU (SHC) - stable harmonic univalent (stable harmonic convex) • PSD (SD) - pre-Schwarzian derivative (Schwarzian derivative) • PSN (SN) - pre-Schwarzian norm (Schwarzian norm) • SBAPSN - stable bounded analytic pre-Schwarzian norm • SBASN - stable bounded analytic Schwarzian norm • SBHPSN - stable bounded harmonic pre-Schwarzian norm • SBHSN - stable bounded harmonic Schwarzian norm • H = { f = h + g : f is a sense-preserving harmonic mapping in D satisfyingthe normalizations h (0) = h ′ (0) − g (0) = 0 }• H = { f = h + g ∈ H : g ′ (0) = 0 } Sometimes we write f ∈ ULU to convey that f is a uniformly locally univalent functionin D . Similar convention will be followed for other cases.1.2. ULU harmonic mappings.
Let z, a ∈ D . We denote the hyperbolic distancebetween z and a by d h ( z, a ) = 2 tanh − (cid:12)(cid:12)(cid:12)(cid:12) z − a − az (cid:12)(cid:12)(cid:12)(cid:12) . The hyperbolic disk in D with center a ∈ D and hyperbolic radius ρ , 0 < ρ ≤ ∞ , isdefined by D h ( a, ρ ) = { z ∈ D : d h ( z, a ) < ρ } . We say that a sense-preserving harmonic mapping f = h + g in D is a ULU harmonicmapping in D if ρ ( f ) >
0, where ρ ( f ) = inf z ∈ D (cid:26) sup ρ z > { ρ z : f is univalent in D h ( z, ρ z ) } (cid:27) . The number ρ ( f ) is called the uniformly hyperbolic radius of f . Moreover, f is univalentin D if and only if ρ ( f ) = ∞ .1.3. PSD and PSN of harmonic mappings.
Let f = h + g be a sense-preservingharmonic mapping in D with ω := ω f = g ′ /h ′ . Then the PSD and the PSN of f aredefined by(1.1) P f = (log J f ) z = h ′′ h ′ − g ′′ g ′ | h ′ | − | g ′ | = h ′′ h ′ − ωω ′ − | ω | armonic mappings, the pre-Schwarzian and the Bloch space 3 and || P f || = sup z ∈ D (1 − | z | ) | P f ( z ) | , respectively. Clearly, in the analytic case ω in (1.1) has taken to be identically 0 in D ,and thus, throughout we use the same notations for the PSD and the PSN in the case ofanalytic functions as well.The PSD has affine invariance property:(1.2) P f = P A ◦ f , A ( z ) = az + bz + c, a, b, c ∈ C and | a | > | b | . Note that A ◦ f is still sense-preserving in D .The above definitions of the PSD and the PSN for harmonic mappings were introducedby Hern´andez and Mart´ın in [21] (see also [11]), which coincide with the correspondinganalytic definitions (see [17, 31]). It is well known that the PSN of a locally univalentanalytic function is an important quantity in the study of the global univalence. Forexample, if f is a univalent analytic function in D , then || P f || ≤
6, which is sharp.Conversely, if || P f || ≤ f in D , then f isnecessarily univalent in D and the constant 1 is sharp (see [7, 8]). Recently, new criteriafor the univalence of harmonic mappings in terms of the PSD or the PSN have beenestablished in [6, 19, 21].1.4. Relationship between ULU and PSN.
Yamashita [40] showed that a locallyunivalent analytic function f in D is ULU in D if and only if || P f || is bounded. Later,Kim and Sugawa [24] investigated the growth of various quantities for a ULU analyticfunction in D by means of finite the PSN. Since P φ ◦ f = P f for any linear transformation φ ( z ) = az + b ( a = 0), they just considered the following normalized function space B A = { f ∈ A : || P f || < ∞} , where A is the set of analytic functions f in D with the normalizations f (0) = f ′ (0) − B A has the structure of a nonseparable complex Banach space underthe Hornich operation (see [39]). To obtain some precise results, it was necessary to studythe subset of B A : B A ( λ ) = { f ∈ A : || P f || ≤ λ } , where λ ≥ f in D is ULU in D if and only if || P f || is bounded, which will be also proved in Section3 by other method. Therefore, the primary aim of this paper is to extend some of theresults from [24] to sense-preserving and ULU harmonic mappings in D associated withfinite the PSN. Since the PSD preserves affine invariance, in what follows, we only toconsider the following set of normalized functions: B H = { f ∈ H : || P f || < ∞} . If we concern only on the PSN, then B H can be further restricted to be B H := B H ∩ H .In fact, if f = h + g ∈ B H and A ( z ) = z − b z −| b | ( b = g ′ (0)), then it follows from (1.2) that || P A ◦ f || = || P f || and it is also easy to see that A ◦ f ∈ B H . G. Liu and S. Ponnusamy
Let f = h + g be a sense-preserving harmonic mapping in D . Then, motivated by theworks of [20], in Section 2, we will build some sharp inequalities between || P h + ε g || and || P h + ε g || , where ε , ε ∈ D . In particular, we obtain the following important implication:(1.3) f ∈ B H ( λ ) := { f ∈ H : || P f || ≤ λ } ⇒ h + εg εg ′ (0) ∈ B A (cid:18) λ + 12 (cid:19) ∀ ε ∈ D , where λ ≥
0. In Section 3, for any given sense-preserving and ULU harmonic mappingin the unit disk, we give a relationship between its PSN and uniformly hyperbolic radius.Combining the above results with some works about ULU harmonic mappings, plentyof equivalent conditions for a sense-preserving and ULU harmonic mapping in the unitdisk are obtained in Section 4. To present some sharp examples in Sections 6 and 7, weintroduce a class of sense-preserving harmonic mappings with prescribed PSN in Section 5.These results help us to obtain sharp distortion, growth and covering theorems for B H ( λ )or B H ( λ ) := B H ( λ ) ∩ H in Section 6. Applying (1.3) and the corresponding results in[24] and [32], the growth of coefficients and the relationship with Hardy space for theclass B H ( λ ) are considered in Sections 7 and 8, respectively. Finally, some subordinationprinciples of the PSN estimates are also obtained in Section 9.2. Some inequalities Concerning Pre-Schwarzian norm
We now state our key inequalities which will provide important connections betweenULU analytic functions and ULU harmonic mappings in the unit disk.
Theorem 2.1.
Let f = h + g be a sense-preserving harmonic mapping in D . Then either || P h + εg || = || P f || = ∞ or both || P h + εg || and || P f || are finite for each ε ∈ D . If || P f || < ∞ ,then the inequality (2.1) (cid:12)(cid:12) || P h + εg || − || P f || (cid:12)(cid:12) ≤ holds for each ε ∈ D . In particular, (cid:12)(cid:12) || P h || − || P f || (cid:12)(cid:12) ≤ . The constant is sharp in the two estimates. Proof.
Suppose that f = h + g is a sense-preserving harmonic mapping in D . Then h + εg is a locally univalent analytic function in D for each ε ∈ D . By (1.1), a direct computationshows that P h + εg = h ′′ + εg ′′ h ′ + εg ′ = P h + εω ′ εω , and thus, P h + εg − P f = εω ′ εω + ωω ′ − | ω | = ε + ω εω · ω ′ − | w | , where ω = g ′ /h ′ . Therefore, by the Schwarz-Pick lemma, we have(1 − | z | ) (cid:12)(cid:12) | P h + εg ( z ) | − | P f ( z ) | (cid:12)(cid:12) ≤ (1 − | z | ) | P h + εg ( z ) − P f ( z ) | ≤ sup z ∈ D (cid:12)(cid:12)(cid:12)(cid:12) ε + ω ( z )1 + εω ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ z ∈ D and the assertion easily follows. armonic mappings, the pre-Schwarzian and the Bloch space 5 To show the sharpness, it suffices to consider the harmonic Koebe function K (see [13])defined by K ( z ) = h ( z ) + g ( z ) = z − z + z (1 − z ) + z + z (1 − z ) , z ∈ D . By a direct computation, we find that P h − g ( z ) = 4 + 2 z − z , P K ( z ) = 5 + 3 z − z − z − | z | ,P h ( z ) = 5 + 3 z − z and P h + g ( z ) = 6 + 2 z − z . It is easy to see that || P h − g || = 6 and || P h || = || P h + g || = 8. Choosing ε = ± || P K || = 7. In summary, we get that || P h − g || + 1 = || P K || = 7 = || P h || − || P h + g || − . In addition, the sharpness can be seen from the harmonic half-plane mapping L (see[13]) defined by L ( z ) = h ( z ) + g ( z ) = 2 z − z − z ) + − z − z ) , z ∈ D . Elementary computations yield that P h + g ( z ) = 21 − z , P L ( z ) = 31 − z − z − | z | ,P h ( z ) = 31 − z and P h − g ( z ) = 4 + 2 z − z . As in the harmonic Koebe function, we obtain that || P h + g || + 1 = || P L || = 5 = || P h || − || P h − g || − (cid:3) Obviously, the assertion (1.3) is true by Theorem 2.1. Next we consider more generalinequalities.
Corollary 2.1.
Let f = h + g be a sense-preserving harmonic mapping in D . If || P h || < ∞ , then for any ε , ε ∈ D , we have the following inequalities. (1) The sharp inequality (cid:12)(cid:12) || P h + ε g || − || P h + ε g || (cid:12)(cid:12) ≤ holds. (2) If | ε | = | ε | , then || P h + ε g || = || P h + ε g || . If | ε | 6 = | ε | , then (cid:12)(cid:12) || P h + ε g || − || P h + ε g || (cid:12)(cid:12) ≤ | ε | + | ε | < . (3) If | ε | ≤ | ε | , then we have the sharp inequality (cid:12)(cid:12) || P h + ε g || − || P h + ε g || (cid:12)(cid:12) ≤ . If | ε | > | ε | , then (cid:12)(cid:12) || P h + ε g || − || P h + ε g || (cid:12)(cid:12) ≤ | ε | + | ε | < . G. Liu and S. Ponnusamy
Proof.
Since || P h || < ∞ , it follows from Theorem 2.1 that both || P h + εg || and || P h + εg || arefinite for each ε ∈ D .(1) The inequality can be easily deduced from (2.1) by applying the triangle inequalityonce. The sharpness can be seen from the harmonic Koebe function and the harmonichalf-plane mapping.(2) Note that f ε = h + εg is still a sense-preserving harmonic mapping with dilatation εω f for any given ε ∈ D . It follows from (1.1) that P h + ε g − P h + ε g = (cid:18) | ε | − | ε ω f | − | ε | − | ε ω f | (cid:19) ω f ω ′ f . Then the former part is trivial. The later part can be easily deduced from the Schwarz-Pick lemma and the triangle inequality.(3) The former part is a direct consequence of (2.1). For the later part, using (2), theformer part and the triangle inequality, we have (cid:12)(cid:12) || P h + ε g || − || P h + ε g || (cid:12)(cid:12) ≤ (cid:12)(cid:12) || P h + ε g || − || P h + ε g || (cid:12)(cid:12) + (cid:12)(cid:12) || P h + ε g || − || P h + ε g || (cid:12)(cid:12) ≤ | ε | + | ε | < . The proof is complete. (cid:3)
Associated with Bieberbach’s criterion and Yamashita’s result about convex analyticfunctions (see [41, Theorem 1]), we get the following result.
Corollary 2.2.
Let f = h + g be a sense-preserving harmonic mapping in D . If h + ε g isunivalent (resp. convex) in D for some ε ∈ D , then || P h + εg || < (resp. ) and || P h + εg || ≤ (resp. ) for each ε ∈ D . Furthermore, the constants and are sharp. Conversely, ifeither || P h + ε g || ≥ (resp. ) or || P h + ε g || > (resp. ) for some ε , ε ∈ D , then h + εg is not univalent (resp. convex) in D for any ε ∈ D . The harmonic Koebe function K = h K + g K and the harmonic half-plane mapping L = h L + g L still show its sharpness in the corresponding cases because h K ( z ) − g K ( z ) = z (1 − z ) is univalent in D and h L ( z ) + g L ( z ) = z − z is univalent and convex in D , respectively.3. Pre-Schwarzian norm and uniformly hyperbolic radius
It is natural to ask whether there exists a generalization of Bieberbach’s criterion forunivalent harmonic mappings. Let S H = ( f ∈ H : f ( z ) = h ( z ) + g ( z ) = ∞ X n =1 a n z n + ∞ X n =1 b n z n is univalent in D ) and S H = S H ∩ H . Set α = sup f ∈S H | a | and α = sup f ∈S H | a | . Clunie and Sheil-Small [13] showed that if f = h + g ∈ S H , then || P h || ≤ α + 1), α < α ≤ α ≤ α + 1 /
2. They conjectured that α ≤ /
2, which has a specialsignificance in many extremal problems for harmonic mappings. The estimate of α wasimproved (see [18, p. 96] and [36, Theorem 10]). Now the best known upper bound for α is in [2]. armonic mappings, the pre-Schwarzian and the Bloch space 7 However, for certain geometric subfamilies of S H , we have some precise coefficient esti-mates. For example, for the families K H and C H of convex and close-to-convex harmonicmappings in D , respectively. We note that K H ⊆ C H ⊆ S H . Set K H = K H ∩ H and C H = C H ∩ H . For these special families, we know (see [13] and [38]):sup f ∈K H | a | = 32 , sup f ∈K H | a | = 2 , sup f ∈C H | a | = 52 and sup f ∈C H | a | = 3 . Therefore, the sharp estimate || P f || ≤ f ∈ K H (see [21, Theorem 4]).On the other hand, based on further research on affine and linear invariant locally univa-lent harmonic mappings, Graf in [19, Theorem 1] obtained that || P f || ≤ f ∈ C H and || P f || ≤ α + 1) for f ∈ S H .In this section, we will first re-certify the above partial results concerning the PSN as adirect consequence of our present study on ULU harmonic mappings. For the convenienceof the reader, we include the proof here since it follows by a direct computation. Notethat the PSN is in general not linear invariant. Theorem 3.1.
Let f = h + g be a sense-preserving and ULU harmonic mapping in D .Then we have (3.1) (1 − | z | ) | P h ( z ) | ≤ α/t + | z | ) and (1 − | z | ) | P f ( z ) | ≤ α /t + | z | ) for every z ∈ D , where t = e ρ ( f ) − e ρ ( f ) + 1 if ρ ( f ) < ∞ , if ρ ( f ) = ∞ .In particular, if f is univalent in D , then || P h || ≤ α + 1) and || P f || ≤ α + 1) . Proof.
Suppose that f = h + g ∈ ULU. Then f is univalent in each hyperbolic disk d h ( z, ρ ( f )) for every z ∈ D . Fix z ∈ D and let φ ( ζ ) = tζ + z tzζ ( ζ ∈ D ), where t is defined asabove. Using the Koebe transformation, we get that F ( ζ ) = ( f ◦ φ )( ζ ) − ( f ◦ φ )(0)( f ◦ φ ) ζ (0)= h ( φ ( ζ )) − h ( z ) th ′ ( z )(1 − | z | ) + g ( φ ( ζ )) − g ( z ) th ′ ( z )(1 − | z | )= H ( ζ ) + G ( ζ )and F ∈ S H . A simple computation yields that | H ′′ (0) | = t (cid:12)(cid:12)(cid:12)(cid:12) (1 − | z | ) h ′′ ( z ) h ′ ( z ) − z (cid:12)(cid:12)(cid:12)(cid:12) ≤ α, G. Liu and S. Ponnusamy which implies the first inequality in (3.1). Using the affine change, we have that F ( ζ ) = F ( ζ ) − b F ( ζ )1 − | b | = H ( ζ ) − b G ( ζ )1 − | b | + G ( ζ ) − b H ( ζ )1 − | b | = H ( ζ ) + G ( ζ )and F ∈ S H , where b = G ′ (0) = g ′ ( z ) /h ′ ( z ). Again, a straightforward computationshows that | H ′′ (0) | = | H ′′ (0) − b G ′′ (0) | − | b | = t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 − | z | ) h ′′ ( z ) h ′ ( z ) − g ′′ ( z ) g ′ ( z ) | h ′ ( z ) | − | g ′ ( z ) | − z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = t | (1 − | z | ) P f ( z ) − z | ≤ α , which implies the second inequality in (3.1). (cid:3) Next we consider stable harmonic univalent (resp. convex) mappings. A sense-preservingharmonic mapping f = h + g in D is called SHU (resp. SHC) if h + λg is univalent (resp.convex) in D for every | λ | = 1. The following result has some similarities with the classicalestimate of the SD for SHU and SHC mappings in [12, Theorem 2], but the method ofproof is different and so can also be adapted to prove [12, Theorem 2]. Theorem 3.2.
Let f = h + g be a sense-preserving harmonic mapping in D . If f is SHU (resp.
SHC ), then we have || P h + εg || ≤ (resp. ) and || P h + εg || ≤ (resp. )for each ε ∈ D . All estimates are sharp. Proof. If f = h + g is SHU (resp. SHC) in D , then both h + εg and h + εg are univalent(resp. convex) in D for each ε ∈ D (see [20]). It follows from Bieberbach’s criterion (resp.[41, Theorem 1]) that || P h + εg || ≤ ε ∈ D .Fix ε ∈ D and let f ε = h + εg . For all z ∈ D , it follows from [21, Lemma 1] that P f ε ( z ) = P h − εω ( z ) g ( z ) and thus,(1 − | z | ) | P f ε ( z ) | = (1 − | z | ) | P h − εω ( z ) g ( z ) | , where ω = g ′ /h ′ . This implies that || P f ε || ≤ sup λ ∈ D || P h + λg || and the assertion follows.To show that all estimates are sharp, it is enough to consider the analytic functions k ( z ) = z (1 − z ) and l ( z ) = 1 + z − z that belong to the families of SHU and SHC mappings with || P k || = 6 and || P l || = 4,respectively. (cid:3) Combining Corollary 2.1 and Theorem 3.1 (resp. Theorem 3.2), we can obtain a fewsimilar results as that of Corollary 2.2 for univalent harmonic mappings (resp. SHU and armonic mappings, the pre-Schwarzian and the Bloch space 9
SHC mappings). However, we do not include these statements here. Below we considerthe converse of Theorem 3.1.
Theorem 3.3.
Let f = h + g be a sense-preserving harmonic mapping in D . If || P f || ≤ M , then f is univalent in the hyperbolic disk D h ( z, t ) for each z ∈ D . Consequently, f is ULU in D and its uniformly hyperbolic radius ρ ( f ) is no less than t . Here t =2 tanh − (1 / (8( M + 1))) . Proof.
Fix ε ∈ D . Let f ε = f + εf = φ ε + ψ ε , where φ ε = h + εg and ψ ε = g + εh . By thehypothesis, (1.2) and (2.1), we get that || P φ ε || ≤ || P f ε || + 1 = || P f || + 1 ≤ M + 1 . It follows from [37, Theorem 2] that φ ε is univalent in D h ( z, t ) for each z ∈ D , where t asthe above. By Hurwtiz’s theorem, we know that for each z ∈ D , h + λg is univalent in D h ( z, t ) for every | λ | = 1. Therefore, it follows from [20, Corollary 2.2] that f is univalentin D h ( z, t ) for each z ∈ D . This ends the proof. (cid:3) Stable geometric properties of ULU analytic and harmonic mappings
In this section, we will show a great number of equivalent conditions for sense-preservingand ULU harmonic mappings in D . First we will introduce some notations. Let f = h + g be a sense-preserving harmonic mapping in D . Set ρ ∗ ( f ) = inf z ∈ D (cid:26) sup ρ z > { ρ z : f is convex in D h ( z, ρ z ) } (cid:27) . If ρ ∗ ( f ) >
0, then we say that f ∈ ULC. The SD and the SN of f were investigated indetails by Hern´andez and Mart´ın [21] (see also [11]) and they were defined by S f = S h + ω − | ω | (cid:18) h ′′ h ′ ω ′ − ω ′′ (cid:19) − (cid:18) ωω ′ − | ω | (cid:19) and || S f || = sup z ∈ D (1 − | z | ) | S f ( z ) | , respectively, where S h is the classical Schwarzian derivative of a locally univalent function h defined by S h = h ′′′ h ′ − (cid:18) h ′′ h ′ (cid:19) . If g is a constant, then it is clear that S f = S h and || S f || = || S h || . Analogous to somefeatures of the PSN, if f is a univalent analytic function in D , we have the sharp inequality || S f || ≤
6. Conversely, if || S f || ≤ f in D , then,according to Krauss-Nehari’s criterion, f is univalent in D and the constant 2 is sharp(see [26, 30]). There are some criteria for the univalence of harmonic mappings in termsof the SN (see [19, 21, 22]), but these results are not sharp.Next, we present equivalent conditions for sense-preserving and ULU harmonic map-pings in D based on the following result. Lemma 4.1. ([17, p. 44] and [40, Theorem 2])
Let f be a locally univalent analyticfunction in D . Then the following are equivalent. (1) f ∈ ULU ; (2) f ∈ ULC ; (3) || P f || < ∞ ; (4) || S f || < ∞ ; (5) There exists a constant m > , and a univalent analytic function F in D such that f ′ = ( F ′ ) m . To describe our results, we introduce the following abbreviations analogous to the paper[20]. Below, let f = h + g be a sense-preserving harmonic mapping in D and ε , ε ∈ D with ε = ε . If h + ε g is ULU (resp. ULC) in D if and only if h + ε g is ULU (resp.ULC) in D , then we say that f is SHULU (resp. SHULC). Similarly, if h + ε g is ULU(resp. ULC) in D if and only if h + ε g is ULU (resp. ULC) in D , then we say that h + g is SAULU (resp. SAULC). If || P h + ε g || (resp. || S h + ε g || ) is bounded if and only if || P h + ε g || (resp. || S h + ε g || ) is bounded, then we say that f has SBHPSN (resp. SBHSN).Similarly, if || P h + ε g || (resp. || S h + ε g || ) is bounded if and only if || P h + ε g || (resp. || S h + ε g || )is bounded, then we say that h + g has SBAPSN (resp. SBASN). Theorem 4.1. (Equivalent conditions)
Let f = h + g be a sense-preserving harmonicmapping in D . Then the following conditions are equivalent. (1) h + g is SAULU ; (2) h + g is SAULC ; (3) h + g has SBAPSN ; (4) h + g has SBASN ; (5) For any two points ε , ε ∈ D with ε = ε , there exists a constant m > , anda univalent analytic function F such that ( h + ε g ) ′ = ( F ′ ) m if and only if thereexists a constant m > , and a univalent analytic function F such that ( h + ε g ) ′ = ( F ′ ) m ;(6) f is SHULU ; (7) f is SHULC ; (8) f has SBHPSN ; (9) f has SBHSN . Proof.
To simplify the proof, we use the equivalent diagram below. If we apply Lemma4.1 to h + ε g and h + ε g , we see that (Ai) and (Bi) ( i = 1 , , ,
4) hold. On the otherhand, (A5), (AB) and (B5) are the direct consequences of Theorem 2.1 and Corollary 2.1.Clearly, the following implications are easy to obtain(1) ⇔ (2) ⇔ (3) ⇔ (4) ⇔ (5) ⇔ (8) . armonic mappings, the pre-Schwarzian and the Bloch space 11 || S h + ε g || < ∞ A ⇐⇒ ( h + ε g ) ′ = ( F ′ ) m ( h + ε g ) ′ = ( F ′ ) m B ⇐⇒ k S h + ε g || < ∞m A m B h + ε g ∈ ULU A ⇐⇒ h + ε g ∈ ULC h + ε g ∈ ULC B ⇐⇒ h + ε g ∈ ULU m A m B || P h + ε g || < ∞ A ⇐⇒ || P h + ε g || < ∞ AB ⇐⇒ || P h + ε g || < ∞ B ⇐⇒ || P h + ε g || < ∞m A m B || S h + ε g || < ∞ A ⇐⇒ h + ε g ∈ ULU h + ε g ∈ ULU B ⇐⇒ || S h + ε g || < ∞m A m B h + ε g ∈ ULC h + ε g ∈ ULC
To complete the proof, we need to show that (6) ⇔ (7) ⇔ (8) ⇔ (9). If we applyTheorems 3.1 and 3.3 to h + ε g and h + ε g , then we obtain the inclusions (A6) and (B6).From [21, Theorem 7], (A7) and (B7) follow.To prove (A8) and (B8), it suffices to show that each f = h + g ∈ ULU also belongsto ULC. To do this, let us assume that f = h + g is ULU in D . Then M = || P f || < ∞ and thus, sup ε ∈ D || P h + εg || ≤ M + 1 by (2.1). Following the proof and notations ofTheorem 3.3, we see that for each z ∈ D , h + λg is convex in D h ( z, (2 − √ t ) forevery | λ | = 1 by the classical result on the radius of convexity (see [17, p. 44]), where t = 2 tanh − (1 / (8( M + 2))). It follows from [20, Theorem 3.1] that f is convex in thehyperbolic disk D h ( z, (2 − √ t ) for each z ∈ D , which means that f is ULC in D .Again, by the bridge (AB), we prove that (6) ⇔ (7) ⇔ (8) ⇔ (9). This completes theproof. (cid:3) Remarks.
In the remarks below, let f = h + g be sense-preserving in D .(1) The pre-Schwarzian norm || P g || and the Schwarzian norm || S g || can be unboundedeven if f and g are univalent and locally univalent in D , respectively. For example,let f n ( z ) = h n ( z ) + g n ( z ) = z − λ ( z − n ( n ≥ < | λ | < / ( n n − )) . It is easy to see that f n is sense-preserving and univalent in D and g n is locallyunivalent in D for any n ≥
2. However, we have that || P g n || = sup z ∈ D (1 − | z | ) n − | − z | = 2( n − → ∞ and || S g n || = sup z ∈ D (1 − | z | ) n − | − z | = 2( n − → ∞ as n → ∞ .(2) On one hand, the dilatation of f can be expressed as square of certain analyticfunction if both h and g are ULU in D . It follows from Lemma 4.1 that || P h || and || P g || are bounded. Let k = max {|| P h || , || P g ||} + 1 and set h ( z ) = Z z ( h ′ ( ζ )) k dζ and g ( z ) = Z z ( g ′ ( ζ )) k dζ in the proof of [40, Theorem 2]. Note that h and g are analytic and univalent in D such that h ′ = ( h ′ ) k and g ′ = ( g ′ ) k . Thus, we have that ω f = g ′ /h ′ = ( g ′ /h ′ ) k .Furthermore, if f is univalent in D , then f can be lifted to a regular minimalsurface given by conformal (or isothermal) parameters in D .(3) On the other hand, the function f , with the dilatation ω f = q , for some analyticfunction q may not belong to ULU. For instance, let f ( z ) = h ( z ) + g ( z ) = e − z +1 z − + z − z − , z ∈ D . A simple computation infers that ω f ( z ) = g ′ ( z ) h ′ ( z ) = (cid:16) e z +1 z − (cid:17) so that | ω f ( z ) | < D and thus, f is sense-preserving in D . However, || P h || = sup z ∈ D (1 − | z | ) | − z || − z | = ∞ , which implies that f is not ULU in D by Theorem 4.1.(4) If the analytic part h is univalent in D , then f is certainly ULU in D by Corollary2.2 and Theorem 4.1. However, the above example shows that even if the co-analytic part g is univalent in D , f may not belong to ULU.5. Some precise examples
In this section, we consider a family of harmonic mappings and compute their PSNsand then discuss the univalency of the corresponding mapping. We next introduce F a,b,θ ( z ) = H a,b ( z ) + G a,b,θ ( z ) , e − iθ G a,b,θ ( z ) := G a,b ( z ) = H a +1 ,b ( z ) − H a,b ( z ) , (5.1)where a, b, θ ∈ R and(5.2) H a,b ( z ) = Z z (1 + t ) a (1 − t ) b dt. If a = b , we denote H a,a by H a . Clearly, H a,b ∈ A and H a,b ( z ) = − H − b, − a ( − z ). Therefore,it is easy to see that F a,b,θ ∈ H with dilatation ω ( z ) = e iθ z and(5.3) F a,b,θ ( z ) = − F − b, − a,θ + π ( − z ) , z ∈ D . In general, computing the PSN and verifying the univalence of a given harmonic mappingare not so easy. Below, we also try to give partial answers to this issue. Moreover, asa byproduct of our investigation, we present some sharp inequalities in Section 6 andgive certain properties of the family B H ( λ ) ( λ ≥ h is a normalized (i.e. h (0) = h ′ (0) − D satisfying the conditionRe (cid:18) zh ′′ ( z ) h ′ ( z ) (cid:19) > − | z | <
1, then h is convex in some direction and hence it is close-to-convex (univalent)in the unit disk. For details and its importance see [34]. armonic mappings, the pre-Schwarzian and the Bloch space 13 Proposition 5.1.
For the functions H a,b and H a defined by (5.2) , we have the followingproperties: (1) k P H a,b k = 2 max {| a | , | b |} . Thus, if max {| a | , | b |} ≤ / , then the functions H a,b areunivalent in D . If max {| a | , | b |} > , then the functions H a,b are not univalent in D . (2) If min {| a | , | b |} + | a − b | ≤ , then the functions H a,b are close-to-convex andunivalent in D . (3) If a ≤ ≤ b ≤ a + 3 , then the functions H a,b are convex in one direction andunivalent in D . Furthermore, if a ≤ ≤ b ≤ a + 2 , then the functions H a,b areconvex in D . (4) The function H a is univalent in D if and only if | a | ≤ . Proof. (1) By computation, for all z ∈ D , we have that(1 − | z | ) | P H a,b ( z ) | = (1 − | z | ) (cid:12)(cid:12)(cid:12)(cid:12) a + b + ( b − a ) z − z (cid:12)(cid:12)(cid:12)(cid:12) ≤ | a + b | + | b − a | = 2 max {| a | , | b |} . Note that lim r → − (1 − r ) | P H a,b ( r ) | = 2 | b | and lim r →− + (1 − r ) | P H a,b ( r ) | = 2 | a | . There-fore, we get that k P H a,b k = 2 max {| a | , | b |} and the result follows by Becker’s univalencecriterion.Note that if max {| a | , | b |} >
3, then k P H a,b k > H a,b can notbe univalent in D .(2) We observe that H ′ a,b ( z ) = (cid:18) z − z (cid:19) a (1 − z ) a − b = (cid:18) z − z (cid:19) b (1 + z ) a − b , and thus, | arg( H ′ a,b ( z )) | < π { ( | a | + | a − b | ) , ( | b | + | a − b | ) } , z ∈ D . If min {| a | , | b |} + | a − b | ≤
1, then we have | arg( H ′ a,b ( z )) | < π in D and thus, by Noshiro-Warschawski’s theorem (see [17]), the functions H a,b are close-to-convex and univalent in D .(3) For a ≤ ≤ b ≤ a + 3, we see thatRe zH ′′ a,b ( z ) H ′ a,b ( z ) ! = 1 + Re az z + Re bz − z > a − b a − b ≥ − z ∈ D and thus, the functions H a,b are convex in one direction and univalent in D .Also, it is clearly that if a ≤ ≤ b ≤ a + 2, thenRe zH ′′ a,b ( z ) H ′ a,b ( z ) ! > , z ∈ D , and thus, the functions H a,b are convex in D .(4) It is a direct consequence of [24, Lemma 2.1] because of H a ( z ) = − H − a ( − z ). (cid:3) Proposition 5.2.
For all θ ∈ R , the family of harmonic mappings F a,b,θ defined by (5.1) has the following properties: (1) || P F a,a,θ || = 2 | a | + 1 = || P H a || + 1 for all a ∈ R . (2) If a ∈ ( −∞ , − ∪ [0 , ∞ ) , then || P F a,a +1 ,θ || = | a + 1 | = || P H a,a +1 || − . However, || P F a,a +1 ,θ || = 1 for each a ∈ ( − , . (3) If a ≤ ≤ b ≤ a + 3 , then the functions F a,b,θ are close-to-convex and univalentin D . (4) The functions F a,a +1 ,θ are univalent in D if and only if a ∈ [ − , . Proof.
By a straightforward computation, we have that P F a,b,θ ( z ) = a + b + ( b − a ) z − z − z − | z | , z ∈ D . (1) It follows from (1.2) and (5.3) that || P F a,a,θ || = || P F − a, − a,θ + π || . So we only need toconsider the case a ≥
0. The conclusion can be easily got by (2.1), Proposition 5.1 andthe fact that || P F a,a,θ || ≥ lim r →− + (1 − r ) | P F a,a,θ ( r ) | = 2 a + 1 = || P H a || + 1 , a ≥ . (2) We first consider the case a ≥
0. Note that h a ( z ) := H a,a +1 ( z ) + e i ( π − θ ) G a,a +1 ,θ ( z ) = H a ( z ) . It follows from (2.1) and Proposition 5.1 that2 a + 1 = || P H a,a +1 || − ≤ || P F a,a +1 ,θ || ≤ || P h a || + 1 = 2 a + 1 , a ≥ . Obviously, || P F a,a +1 ,θ || = 2 a + 1 = || P H a,a +1 || − a ≥ θ ∈ R . For thecase a ≤ −
1, the conclusion follows, since || P F a,a +1 ,θ || = || P F − ( a +1) , − ( a +1)+1 ,θ + π || .Next we will certify that || P F a,a +1 ,θ || = 1 for each a ∈ ( − , | − z | − ((2 a + 1) (1 − | z | ) − ( z − z ) ) = − a ( a + 1)(1 − | z | ) > , z ∈ D , which means that(1 − | z | ) | P F a,a +1 ,θ ( z ) | = (cid:12)(cid:12)(cid:12) (2 a + 1)(1 − | z | ) + z − z − z (cid:12)(cid:12)(cid:12) ≤ , z ∈ D . It yields that || P F a,a +1 ,θ || ≤
1. Note that lim r → − (1 − r ) | P F a,a +1 ,θ ( ir ) | = 1 and thus, weobtain || P F a,a +1 ,θ || = 1.(3) If a ≤ ≤ b ≤ a + 3, then from the proof of Proposition 5.1 we find thatRe zH ′′ a,b ( z ) H ′ a,b ( z ) ! > − , z ∈ D . Note that the dilatation of F a,b,θ is e iθ z for all a, b ∈ R and each θ ∈ R . As a consequence,it follows from [9, Theorem 1] that the functions F a,b,θ are close-to-convex and univalentfor all θ ∈ R if a ≤ ≤ b ≤ a + 3.(4) Obviously, it follows from (3) that the functions F a,a +1 ,θ are univalent in D forall θ ∈ R if a ∈ [ − , a > F a,a +1 ,θ are univalent in D for all θ ∈ R . Therefore, the function F a,a +1 ,θ is stable univalent for each θ ∈ R and thus H a,a +1 + λG a,a +1 ,θ is univalent in D for each λ ∈ D (see [20]), especially for λ = e − iθ . armonic mappings, the pre-Schwarzian and the Bloch space 15 However, H a,a +1 + e − iθ G a,a +1 ,θ = H a +1 ,a +1 is not univalent in D by Proposition 5.1 when a >
0. This is a contradiction. Using (5.3), the similar contradiction can be obtained forthe case a < −
1. This completes the proof. (cid:3)
From the proof of Proposition 5.2, the two families of harmonic mappings F a,a,θ and F a,a +1 ,θ provide sharp results for several of the inequalities in Section 2. For simplicity,let h a,b,θ,ϕ ( z ) = H a,b ( z ) + e iϕ G a,b,θ ( z ) = H a,b ( z ) + e i ( θ + ϕ ) ( H a +1 ,b ( z ) − H a,b ( z )) . Clearly, h a,b,θ, − θ = H a +1 ,b and h a,b,θ,π − θ = 2 H a,b − H a +1 ,b = H a,b − . We have the followingresults from Propositions 5.1 and 5.2. • || P h a,a,θ,π − θ || + 1 = || P H a || + 1 = || P F a,a,θ || = 2 a + 1 = || P h a,a,θ, − θ || − a ≥ / • || P H a || + 1 = || P F a,a,θ || = 2 a + 1 = || P h a,a,θ, − θ || −
1, 0 ≤ a ≤ / • || P F a,a,θ || = 2 a + 1 = || P h a,a,θ,π − θ || + ε , a = (1 + ε ) / ∈ [1 / , / • || P F a,a,θ || = 2 a + 1 = || P h a,a,θ,π − θ || − ε , a = (1 − ε ) / ∈ [0 , / • || P h a,a +1 ,θ,π − θ || + 1 = || P F a,a +1 ,θ || = 2 a + 1 = || P H a,a +1 || − || P h a,a +1 ,θ, − θ || − a ≥ F a,a,θ and F a,a +1 ,θ when a <
0. For these functions, weknow that || P F a,b,θ || ≥ ω F a,b,θ = e iθ z for all a , b = a or a + 1, θ ∈ R . These thingsdo not happen accidentally. Our next result, which is a parallel result to [12, Theorem 3],demonstrates the reason behind these.We denote by A ( λ ) (resp. A ( λ )) the set of all admissible dilatations of f ∈ B H ( λ ) (resp. B H ( λ )); i.e., ω ∈ A ( λ ) (or A ( λ )) if there exists a harmonic mapping f = h + g ∈ B H ( λ )( B H ( λ )) with dilatation ω . Theorem 5.1.
The following conditions are equivalent. (1) λ ≥ ; (2) There exists a ω ∈ A ( λ ) with | ω ′ (0) | = 1 ; (3) The set { µ · I : | µ | = 1 } is contained in A ( λ ) ; (4) Every automorphism σ of the unit disk is an admissible dilatation in B H ( λ ) . Proof.
The scheme of the proof is to show that (1) ⇒ (2) ⇒ (3) ⇒ (1) and (3) ⇔ (4).We only show (1) ⇒ (2) and (3) ⇒ (1). The remaining implications are similar tocorresponding proofs of [12, Theorem 3] since the PSD preserves affine invariance.We now show that (1) ⇒ (2): For any given λ ≥
1, we choose | a | = λ − so that P F a,a,θ ∈ B H ( λ ) with the dilatation e iθ z by Proposition 5.2. Then (2) follows.Next, we prove that (3) ⇒ (1): If (3) is satisfied, then for any given µ with | µ | = 1,there is a harmonic function f µ = h µ + g µ ∈ B H ( λ ) with dilatation µz . Since g ′ µ (0) = 0,by (1.3), h µ + εg µ ∈ B A (( λ + 1) /
2) for each ε ∈ D . It follows from [24, Theorem 2.3] thatfor any ε ∈ D ,(5.4) | ( h ′ µ + εg ′ µ )( z ) | = | h ′ µ ( z ) | · | εµz | ≥ (cid:18) − | z | | z | (cid:19) λ +12 , z ∈ D . Since ε ∈ D and | µ | = 1, for any given z = 0 in the unit disk, we can get | h ′ µ ( z ) | ≥ (1 − | z | ) λ − (1 + | z | ) λ +12 , by choosing ε · µ = − z/ | z | in (5.4). Clearly, the above inequality holds for z = 0. Notethat h µ is locally univalent in D and h ′ µ (0) = 1 for each µ ∈ ∂ D . We obtain that theanalytic function 1 /h ′ µ satisfies 1 | h ′ µ ( z ) | ≤ (1 + | z | ) λ +12 (1 − | z | ) λ − , z ∈ D , which implies that λ ≥ / | h ′ µ (0) | <
1, which contradicts h ′ µ (0) = 1. This completes the proof. (cid:3) Compared to [12, Theorem 3], since the PSD is in general not linear invariant, weare not sure whether the conditions in Theorem 5.1 are equivalent to that there exists ω ∈ A ( λ ) (or A ( λ )) with || ω ∗ || = 1. Here || ω ∗ || is the hyperbolic norm of the dilatation ω of a sense-preserving harmonic mapping in D , i.e., || ω ∗ || = sup z ∈ D | ω ′ ( z ) | (1 − | z | )1 − | ω ( z ) | . The hyperbolic norm plays a distinguished role in the analysis of the order of affine andlinear invariant families of harmonic mappings with bounded SD (see [12]).6.
Growth estimate for the class B H ( λ )To study the growth estimate for the class B H ( λ ), we need the following result whichcharacterizes harmonic mappings in B H ( λ ). Proposition 6.1.
A harmonic mapping f ∈ H belongs to B H ( λ ) if and only if for eachpair of points z , z in D , the inequality | A ( z ) − A ( z ) | ≤ λd h ( z, z ) holds, where A ( z ) = log J f ( z ) . Proof.
Assume that f = h + g ∈ B H ( λ ). Then | P f ( z ) | ≤ λ/ (1 − | z | ) holds in D . Weobserve that A z = (log J f ) z = (log J f ) z = P f = A z . Therefore, for two points z , z in D , we have that | A ( z ) − A ( z ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z Γ A ζ ( ζ ) dζ + A ζ ( ζ ) dζ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Γ ( | A ζ ( ζ ) | + | A ζ ( ζ ) | ) | dζ |≤ Z Γ λ − | ζ | | dζ | = λd h ( z, z ) , where Γ is the hyperbolic geodesic joining z and z . armonic mappings, the pre-Schwarzian and the Bloch space 17 Conversely, we assume that the inequality | A ( z ) − A ( z ) | ≤ λd h ( z , z ) holds for eachpair of points z , z in D . It suffices to prove that (1 − | z | ) | P f ( z ) | ≤ λ for each z ∈ D .Fix z ∈ D . If A z ( z ) = 0, then P f ( z ) = 0 and (1 − | z | ) | P f ( z ) | ≤ λ . Otherwise, choosea curve γ = { z = z + re − iθ : r ∈ (0 , − | z | ) , θ = arg( A z ( z )) } . Clearly, A ( z ) is infinitely differentiable in D owing to J f ( z ) >
0. Thus, we have therepresentation A ( z ) − A ( z ) = A z ( z )( z − z ) + A z ( z )( z − z ) + X i + j> C ij ( z − z ) i ( z − z ) j for some complex constants C ij , which implies thatlim γ ∋ z → z A ( z ) − A ( z ) z − z = A z ( z ) + A z ( z ) e iθ = 2 A z ( z ) = 2 P f ( z ) . The desired inequality (1 − | z | ) | P f ( z ) | ≤ λ follows from the equalitylim γ ∋ z → z | A ( z ) − A ( z ) | d h ( z, z ) = (1 − | z | ) | P f ( z ) | . The proof is complete. (cid:3)
Theorem 6.1. (Distortion theorem)
Let f = h + g ∈ B H ( λ ) for some λ ≥ with b = g ′ (0) , and let H a,b and H a be defined by (5.2) . Then for each z ∈ D , we have (1) (1 − | b | ) H ′ λ ( −| z | ) ≤ J f ( z ) ≤ (1 − | b | ) H ′ λ ( | z | ) ; (2) p − | b | H ′ λ ( −| z | ) ≤ | h ′ ( z ) | ≤ (1 + | b z | ) H ′ λ − , λ +12 ( | z | ) ; (3) | g ′ ( z ) | ≤ ( | z | + | b | ) H ′ λ − , λ +12 ( | z | ) ; (4) − p − | b | H λ ( −| z | ) ≤ | h ( z ) | ≤ (1 − | b | ) H λ − , λ +12 ( | z | ) + | b | H λ +12 ( | z | ) ; (5) | g ( z ) | ≤ H λ +12 ( | z | ) − (1 − | b | ) H λ − , λ +12 ( | z | ) ; (6) | f ( z ) | ≤ (1 + | b | ) H λ +12 ( | z | ) . Furthermore, if f is univalent in D , then − (1 − | b | ) H λ +12 ( −| z | ) ≤ | f ( z ) | ≤ (1 + | b | ) H λ +12 ( | z | ) . The estimates in (1) are sharp for all λ ≥ . The right sides of (2)-(6) are sharp forall λ ≥ and the left side of (6) is sharp for λ = 1 . Moreover, if f ∈ B H ( λ ) , then the leftsides of (2) and (4) are sharp for all λ ≥ . Proof. (1) The conclusion can be easily obtained by choosing z = 0 in Proposition 6.1.(2) Since f ∈ B H ( λ ), by Lindel¨of’s inequality, we get that | ω f ( z ) | ≤ | z | + | b | | b z | and thus, | h ′ ( z ) | = (cid:18) J f ( z )1 − | w f ( z ) | (cid:19) ≤ (1 − | b | ) − (cid:18) | z | + | b | | b z | (cid:19) ! − (cid:18) | z | − | z | (cid:19) λ = (1 + | b z | ) H ′ λ − , λ +12 ( | z | )and | h ′ ( z ) | ≥ ( J f ( z )) ≥ p − | b | (cid:18) − | z | | z | (cid:19) λ = p − | b | H ′ λ ( −| z | ) . (3) It follows from Lindel¨of’s inequality and the proof of (2) that | g ′ ( z ) | = | w f ( z ) h ′ ( z ) | ≤ ( | z | + | b | ) H ′ λ − , λ +12 ( | z | ) . (4) Integrating inequalities in (2) yields (4).(5) Integrating inequality in (3) yields (5).(6) Applying the triangle inequality and the results in (4) and (5), we obtain | f ( z ) | ≤ | h ( z ) | + | g ( z ) | ≤ (1 + | b | ) H λ +12 ( z ) . Let f ε = h + εg εb ( ε ∈ D ). Then f ε belongs to B A ( λ +12 ). By [24, Theorem 2.3], we have | ( h ′ + εg ′ )( z ) | ≥ | εb | (cid:18) − | z | | z | (cid:19) λ +12 ≥ (1 − | b | ) H ′ λ +12 ( −| z | ) . Especially, since ε is arbitrary, we get that | h ′ ( z ) | − | g ′ ( z ) | ≥ (1 − | b | ) H ′ λ +12 ( −| z | ) . For 0 < r < z such that | f ( z ) | is the minimum of | f ( z ) | on | z | = r . If f isunivalent in D and γ is the preimage of the segment [0 , f ( z )], then for | z | = r , we havethat | f ( z ) | ≥ | f ( z ) | = Z γ | df ( z ) | ≥ Z | z | ( | f z ( z ) | − | f z ( z ) | ) | dz | ≥ − (1 − | b | ) H λ +12 ( −| z | ) . Next we consider the sharpness part. The equality occurs in (1) if we take f ( z ) = H λ ( z ) + b H λ ( z ) for each λ ≥ . For each λ ≥
1, the equalities in the right sides of (2)-(6) are attained for f ( z ) = f λ ( z ) = F λ − , λ +12 , ( z ) + | b | F λ − , λ +12 , ( z )at z = r ∈ [0 , F a,b, is defined by (5.1). Note that f λ ∈ B H ( λ ) by (1.2) andProposition 5.2. Similarly, for each λ ≥
0, the function H λ provides the sharpness for theleft sides of (2) and (4) at z = − r ∈ ( − ,
0] when f ∈ B H ( λ ). It follows from Proposition5.2 that F λ − , λ +12 , is univalent in D for λ = 1. The equality in the left side of (6) occursfor f = F , , − | b | F , , ∈ B H (1) and z = − r ∈ ( − , (cid:3) armonic mappings, the pre-Schwarzian and the Bloch space 19 The following result can be directly deduced from Theorem 6.1.
Corollary 6.1. (Growth and covering theorem)
Let f = h + g ∈ B H ( λ ) with b = g ′ (0) ,and let H a,b and H a be defined by (5.2) . If λ > , then f , h and g satisfy the same growthcondition f ( z ) ( h ( z ) , g ( z )) = O (1 − | z | ) − λ as | z | → .If λ < , then f (resp. h , g ) is bounded by (1+ | b | ) H λ +12 (1) ( resp. (1 −| b | ) H λ − , λ +12 (1)+ | b | H λ +12 (1) , H λ +12 (1) − (1 −| b | ) H λ − , λ +12 (1)) . For all λ > , the image h ( D ) contains the disk {| z | < − p − | b | H λ ( − } . If f ∈ B H ( λ ) ∩ S H , then the image f ( D ) contains the disk {| z | < − (1 − | b | ) H λ +12 ( − } . If f ∈ B H ( λ ) ∩ S H for some λ ∈ [0 , − H λ +12 ( − ≥ − H ( −
1) = 2 log 2 − . · · · . This result is an improvement over the non-sharp known result that f ( D ) ⊇ { w : | w | < / } if f ∈ S H .In Corollary 6.1, the case λ = 1 is critical. By Theorem 6.1, we have that, for f ∈ B H (1), | f ( z ) | ≤ (1 + | b | ) H ( | z | ) = (1 + | b | ) (cid:0) − − | z | ) − | z | (cid:1) , z ∈ D , which shows that functions in B H (1) need not be bounded. The following result gives asufficient condition for the boundedness of mappings in B H (1). Proposition 6.2.
Let f = h + g be a sense-preserving harmonic mapping in D . If f satisfies the condition β ( f ) := lim | z |→ − ((1 − | z | ) | P f ( z ) | −
1) log 11 − | z | < − , then f , h and g are bounded in D . Proof.
Without loss of generality, we can assume that g (0) = 0. It follows from (1.2) that β ( A ◦ f ) = β ( f ) for any affine harmonic mapping A defined in (1.2). Let A ◦ f = H + G .It is easy to check that both h and g are bounded in D if and only if both H and G are bounded in D . Note that A ◦ f is also sense-preserving in D . Thus, it is enough toconsider the case f = h + g ∈ H and prove that both h and g are bounded in D .By assumption, there exist β < − r ∈ (1 − / (2 e ) ,
1) such that(6.1) | P f ( z ) | ≤ − | z | + β (1 − | z | ) log(1 / − | z | )for z ∈ D r = { z : r < | z | < } . Fix z ∈ D r and let Γ be a line segment from z to z := r e i arg z in the proof of Proposition 6.1. Then we have(6.2) | log J f ( z ) | ≤ Z | z | r | P f ( ζ ) || dζ | + C , where C = max θ ∈ [0 , π ] | J f ( r e iθ ) | < ∞ . By (6.1) and (6.2), we see that | log J f ( z ) | ≤ log 1 + | z | − | z | + Z | z | r βdt (1 − t ) log(1 / (1 − t )) + C ≤ log 1 + | z | − | z | + Z | z | r βdt (1 − t ) log(1 / (2(1 − t ))) + C = log 1 + | z | − | z | + β log log 12(1 − | z | ) + C , where C = C − β log log − r ) . Exponentiating the last inequality shows that | J f ( z ) | = | h ′ ( z ) | (1 − | ω f ( z ) | ) ≤ e C | z | − | z | (cid:18) log 12(1 − | z | ) (cid:19) β . Using that f ∈ B H (1), we have | ω f ( z ) | ≤ | z | in D and thus, we find that | g ′ ( z ) | < | h ′ ( z ) | = (cid:18) J f ( z )1 − | ω f ( z ) | (cid:19) / ≤ e C / − | z | (cid:18) log 12(1 − | z | ) (cid:19) β/ . Since β/ < −
1, the function [log(1 / (2(1 − t )))] β/ / (1 − t ) is integrable on the interval[ r , h and g are bounded in D so that f is also bounded in D . (cid:3) Remark.
Let f = h + g be a sense-preserving harmonic mapping in D . If h and g areunbounded in D , then the boundedness of f is uncertain. For instance, let’s recall thefunction F , ,θ defined by (5.1): F , ,θ ( z ) = H , ( z ) + G , ,θ ( z ) = − log(1 − z ) + e iθ ( − z − log(1 − z )) , we see that H , and G , ,θ are unbounded in D . However, F , , ( z ) = − z − | − z | and F , ,π ( z ) = z − − z ) are unbounded and bounded in D , respectively. Furthermore,it follows from Proposition 5.2 that || P F , ,θ || = 1 for any θ ∈ R .By Theorem 6.1 and [16, Theorem 5.1], we conclude the H¨older continuity of mappingsin B H ( λ ). Theorem 6.2.
Let f = h + g ∈ B H ( λ ) for some λ ∈ [0 , . Then h + εg is H¨oldercontinuous of exponent − λ in D for each ε ∈ D . Moreover, f is H¨older continuous ofexponent − λ in D . Coefficient estimates for the class B H ( λ )Throughout the section we consider f = h + g ∈ B H , where h ( z ) = ∞ X n =1 a n z n and g ( z ) = ∞ X n =1 b n z n with a = 1 and B H is defined in Section 1.4. For ε ∈ D , we now introduce f ε by f ε ( z ) := h ( z ) + εg ( z )1 + εb = 11 + εb ∞ X n =1 ( a n + εb n ) z n . armonic mappings, the pre-Schwarzian and the Bloch space 21 We first determine the estimate for a . Theorem 7.1. If f ∈ B H ( λ ) , then we have (7.1) | a | ≤
12 min (cid:26) (1 − | b | ) λ + 2 | b b | , min ε ∈ D {| εb | ( λ + 1) + 2 | εb |} (cid:27) . If f ∈ B H ( λ ) , then | a | ≤ λ/ and the estimate is sharp for all λ > . Proof.
Let ε ∈ D and f ∈ B H ( λ ) for some λ >
0. Then f ε defined above belongs to B A (( λ + 1) / | P f ε (0) | ≤ || P f ε || ≤ λ + 1 so that (cid:12)(cid:12)(cid:12)(cid:12) h ′′ (0) + εg ′′ (0) h ′ (0) + εg ′ (0) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) a + 2 εb εb (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ + 1 , which implies that | a | ≤ ( | εb | ( λ + 1) + 2 | εb | ).On the other hand, for f ∈ B H ( λ ), it follows from (1.1) that | P f (0) | = | h ′′ (0) h ′ (0) − g ′′ (0) g ′ (0) || h ′ (0) | − | g ′ (0) | = | a − b b | − | b | ≤ || P f || = λ, and thus, | a | ≤ ((1 − | b | ) λ + 2 | b b | ). Inequality (7.1) follows if we combine the lasttwo estimates for | a | . If f ∈ B H ( λ ), then b = 0 and thus, (7.1) reduces to | a | ≤ λ/ H λ/ = H λ/ ,λ/ defined by (5.2) provides the sharpness for each λ > (cid:3) In order to indicate estimates for the coefficients of f ∈ B H ( λ ), we consider the integralmean I p ( r, f ) of f defined by I p ( r, f ) = 12 π Z π | f ( re iθ ) | p dθ, where p is a positive real number. Set M p ( r, f ) = ( I p ( r, f )) /p , 0 < r < Definition 7.1.
For < p < ∞ , the Hardy space H p is the set of all functions f analyticin D for which k f k p := sup { M p ( r, f ) : 0 < r < } < + ∞ , where M p ( r, f ) is defined asabove.Let h p denote the analogous space of harmonic mappings f in D with k f k p definedsimilarly (see [18] ). In [4], Aleman and Mart´ın constructed convex harmonic mappings that do not belongto h / which settles the question raised by Duren [18]. It is worth pointing out that thespace h p is well-behaved for p ≥ H p is comparatively well-behaved for all p > h p , 0 < p < f ∈ B H ( λ ) implies that f ε ∈ B A (( λ + 1) /
2) for each ε ∈ D , it follows from theresult of [24, p. 190] that | a n + εb n | = O (cid:0) n ( λ +1) / − (cid:1) uniformly for ε ∈ D as n → ∞ and thus, we obtain that | a n | + | b n | = O (cid:0) n ( λ − / (cid:1) as n → ∞ . Especially, | a n | = O (cid:0) n ( λ − / (cid:1) and | b n | = O (cid:0) n ( λ − / (cid:1) as n → ∞ . Moreover, if λ < f is univalent in D , then, by Corollary 6.1, f is bounded and thus,Area ( f ( D )) = π (cid:16) ∞ X n =1 n ( | a n | − | b n | ) (cid:17) < ∞ , which implies that p | a n | − | b n | = o (cid:0) n − / (cid:1) as n → ∞ .Combining the results from [24, Section 3] and the implication (1.3), we can get a seriesof results. We omit detailed proofs, but it might be appropriate to include some necessaryexplanations. In fact, we only need to modify the conditions by replacing the parameter λ in the theorems of [24, Section 3] by ( λ + 1) / Theorem 7.2.
Let f = h + g ∈ B H ( λ ) . Then, for any a > and a real number p , wehave (7.2) I p ( r, h ′ + εg ′ ) = O (cid:0) (1 − r ) − α ( | p | ( λ +1) / − a (cid:1) , for each ε ∈ D and thus, in particular, | a n | + | b n | = O (cid:0) n α (( λ +1) / − a (cid:1) . For p > , we get that (7.3) I p ( r, f ) = O (cid:0) (1 − r ) p − α ( | p | ( λ +1) / − a (cid:1) . Here α ( λ ) = √ λ − . Proof.
The former part can be deduced from [24, Theorem 3.1]. For the later part, itfollows from (7.2) and [16, Theorem 5.5] that M p ( r, h ′ ) = O (cid:0) (1 − r ) − ( α ( | p | ( λ +1) / a ) /p (cid:1) and M p ( r, h ) = O (cid:0) (1 − r ) − ( α ( | p | ( λ +1) / a ) /p (cid:1) , respectively. Similar conclusions hold for g . Because M p ( r, f ) ≤ p ( M p ( r, h ) + M p ( r, g )),we finally obtain that M p ( r, f ) = O (cid:0) (1 − r ) − ( α ( | p | ( λ +1) / a ) /p (cid:1) , which implies (7.3). (cid:3) Theorem 7.3.
Let f = h + g ∈ B H ( λ ) for some λ with . < λ ≤ . If there exists aconstant ε ∈ D such that h + εg is univalent in D , then | a n + εb n | = O (cid:0) n ( λ − / (cid:1) as n → ∞ .In particular, if h is univalent in D , then | a n | = O (cid:0) n ( λ − / (cid:1) as n → ∞ . Moreover, if h + εg is univalent in D for every | ε | = 1 , then | a n | + | b n | = O (cid:0) n ( λ − / (cid:1) as n → ∞ . Thethree estimates are sharp. Proof. If f ∈ B H ( λ ) for some λ with 1 . < λ ≤
5, then as before we have f ε ∈ B A ( λ +12 )(1 . < λ +12 ≤
3) for each ε ∈ D . The results follow from [24, Theorem 3.2].To show the sharpness, we construct a family of functions T λ,θ ( z ) = t λ ( z ) + e iθ zt λ ( z ) = ∞ X n =1 a n z n + ∞ X n =2 b n z n , z ∈ D , where λ ∈ (1 . , θ ∈ R and t λ ( z ) = 1 − (1 − z ) (1 − λ ) / (1 − λ ) / . armonic mappings, the pre-Schwarzian and the Bloch space 23 First, we show that T λ,θ ∈ B H ( λ ) for all λ > θ ∈ R . It suffices to prove that || P T λ,θ || = λ due to T λ,θ ∈ H . By computation, we find that P T λ,θ ( z ) = 1 + λ · − z − z − | z | . Also, we note that || P t λ || = 1 + λ . If we get || P T λ,θ || ≤ λ , then it follows from (2.1) that || P T λ,θ || = λ . Indeed we may let z = x + iy ∈ D . By computation, we obtain4 λ | − z | − (cid:12)(cid:12) (1 + λ )(1 − | z | ) − z (1 − z ) (cid:12)(cid:12) =( λ − − x ) (1 − x + λ (3 + x )) + 2(3 + 3 λ − x + (1 − λ ) x ) y − ( λ − y ] ≥ ( λ − λ − x + (1 − λ ) x ) y − λ − y ] ≥ λ − λ − − λ − λ + 1) y ≥ , which clearly implies that (1 − | z | ) | P T λ,θ ( z ) | ≤ λ and thus, || P T λ,θ || ≤ λ . Next, we show that the functions T λ,θ are univalent in D for each1 < λ ≤ θ ∈ R . A simple computation shows thatRe (cid:18) z t ′′ λ ( z ) t ′ λ ( z ) (cid:19) = Re (cid:18) λ · z − z (cid:19) > − λ ≥ − , z ∈ D , for 1 < λ ≤
5. According to a well-known result, the function t λ is univalent and convexin one direction (and hence, close-to-convex) in D . Note that the dilatation of T λ,θ is e iθ z . It follows from [9, Theorem 1] that the functions T λ,θ are univalent in D for each1 < λ ≤ θ ∈ R . Therefore, T λ,θ is SHU and thus, t λ + εzt λ is univalent in D foreach ε ∈ D . Finally, by Stirling’s formula, we have | a n | = 2Γ(( λ + 2 n − / λ − n !Γ(( λ − / ∼ λ − n ( λ − / as n → ∞ . Note that b n = e iθ a n − for each n >
1. Hence, | a n | + | b n | = | a n | + | a n − | = O (cid:0) n ( λ − / (cid:1) as n → ∞ . (cid:3) Given a harmonic mapping f ∈ H , let γ ( f ) denote the infimum of exponents γ suchthat | a n | + | b n | = O (cid:0) n γ − (cid:1) as n → ∞ , that is, γ ( f ) = lim n →∞ log n ( | a n | + | b n | )log n . For the subset X of H , we let γ ( X ) = sup f ∈ X γ ( f ). There are some investigations about γ ( f ) (resp. γ ( X )) if f (resp. X ) is restricted to be analytic or special families of analyticfunctions. The reader can refer to [10, 24, 28] and [31, Chapter 10] for some details onthis problem.For each λ ∈ (0 , ∞ ) and ε ∈ D , we introduce A H ( λ, ε ) =: (cid:26) f ε : f ε ( z ) = h ( z ) + εg ( z )1 + εg ′ (0) and f = h + g ∈ B H ( λ ) (cid:27) and obtain the following theorem. Theorem 7.4.
For each λ ∈ (0 , ∞ ) and ε ∈ D , we have max { ( λ − / , } ≤ γ ( B H ( λ )) ( γ ( A H ( λ, ε ))) ≤ α (( λ + 1) / , where α ( λ ) = √ λ − . In particular, γ ( B H ( λ )) = O (( λ + 1) ) and γ ( A H ( λ, ε )) = O (( λ + 1) ) as λ → . We continue the discussion by mentioning a connection with integral means for univalentanalytic functions. For a univalent harmonic mapping f = h + g ∈ S H , a complex number ε ∈ D and a real number p , we let β f ε ( p ) = lim r → − log I p ( r, f ′ ε )log − r . Clearly, for a univalent analytic function f ∈ A ∩ S H , β f ( p ) = lim r → − log I p ( r, f ′ )log − r . Brennan conjectured that β f ( − ≤ f (see [31, Charpter 8]).As a corollary to Theorem 7.2, we have Theorem 7.5.
For f ∈ B H ( λ ) and a real number p , β f ε ( p ) ≤ α ( | p | ( λ + 1) /
2) = p p (1 + λ ) − holds for each ε ∈ D . In particular, the Brennan conjecture is true for every univalentharmonic mapping f with k P f k ≤ √ − . The space B H ( λ ) and the Hardy space For a harmonic mapping f = h + g in D , the Bloch seminorm is given by (see [14]) k f k B H = sup z ∈ D (1 − | z | ) (cid:0) | h ′ ( z ) | + | g ′ ( z ) | (cid:1) , and f is called a (harmonic) Bloch mapping when k f k B H < ∞ . Let BMOA (resp. BMOH)denote the class of analytic functions (resp. harmonic mappings) that have bounded meanoscillation on the unit disk D (see [1]). Kim [23] showed some relationships among B A ( λ ), H p and BMOA (see also [25]). Combined with the study on Bloch, BMO and univalentharmonic mappings (see [1]), a generalization of Kim’s result is given in [32]. Basicproperties about analytic Bloch functions may be obtained from [5, 31].Our results are based on the following observation. It follows from Theorem 6.1 (6)that the inequality | f ( z ) | ≤ (1 + | b | ) Z | z | (cid:18) t − t (cid:19) λ +12 dt, z ∈ D , holds for every f ∈ B H ( λ ), which implies that • f is bounded when λ < • f ( z ) = O ( − log(1 − | z | )) ( | z | →
1) when λ = 1, and • f ( z ) = O ((1 − | z | ) − λ +12 ) ( | z | →
1) when λ > armonic mappings, the pre-Schwarzian and the Bloch space 25 On the other hand, the proofs of our results are similar to that of results of [32, Sec-tion 4]. Let T H ( λ ) = { f = h + g ∈ H : k T f k ≤ λ } with k T f k := sup z ∈ D , θ ∈ [0 , π ] (1 − | z | ) (cid:12)(cid:12)(cid:12)(cid:12) h ′′ ( z ) + e iθ g ′′ ( z ) h ′ ( z ) + e iθ g ′ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . For the one parameter family T H ( λ ), the authors showed its relationship with Hardyspaces in [32, Section 4]. Note that || T f || = sup θ ∈ [0 , π ] || P h + e iθ g || . If f ∈ B H ( λ ), then itis easy to see that f ∈ T H ( λ +12 ) from (2.1). Therefore, applying the above observationand replacing h + e iθ g ( θ ∈ [0 , π ]) (resp. λ ) to h + εg ( ε ∈ D ) (resp. ( λ + 1) /
2) incorresponding proofs of [32, Section 4], then we can easily obtain the following results.So we omit their proofs.
Theorem 8.1. (1) If λ < , then B H ( λ ) ∩ S H ⊂ h ∞ . (2) If λ = 1 , then B H ( λ ) ∩ S H ⊂ BMOH . (3) If λ > , then B H ( λ ) ∩ S H K ⊂ h p for every < p < / ( λ − , where K ≥ and S H K = { f = h + g ∈ S H : f is K -quasiconformal } . Theorem 8.2.
Let λ ≥ . Then B H ( λ ) ⊂ h p with < p < p ( λ ) = λ +3)( λ − , where p ( λ ) = ∞ if λ = 1 . Remark.
Theorem 8.2 can be directly obtained by choosing p − α ( p ( λ + 1) / > Corollary 8.1.
A uniformly locally univalent harmonic mapping f in D is contained inthe Hardy space h p for some p = p ( f ) > . Subordination principles for the estimate of PSN
In this section, A D denotes the class of analytic functions φ from D into itself and A D denotes the subclass of A D with the normalization φ (0) = 0. If f and F are restricted tobe analytic, then we say that f is said to be subordinate (resp. weakly subordinate ) to F (written f ≺ F (resp. f (cid:22) F )) if there exists a function φ ∈ A D (resp. φ ∈ A D ) suchthat f ( z ) = F ( φ ( z )) in D .In 2000, Schaubroeck [35] generalized the notion of subordination to harmonic map-pings. A harmonic mapping f is subordinate to a harmonic mapping F , still denoted by f ≺ F , if there is a function φ ∈ A D such that f = F ◦ φ .Note that if the analytic function F is univalent in D , then f ≺ F if and only ifthat f (0) = F (0) and f ( D ) ⊆ F ( D ). However, this property is not true for harmonicmappings. As in [29], a harmonic mapping f is said to be weakly subordinate to theharmonic mapping F if f ( D ) ⊆ F ( D ).In this article, f = h + g (cid:22) F = H + G means that there exists a function φ ∈ A D such that h = H ◦ φ and g = G ◦ φ . Clearly, if f (cid:22) F , then f is weakly subordinate to F in the sense of Muir. The following result is a generalization of [24, Theorem 4.1]. Theorem 9.1. (Subordination principle I)
Let f = h + g be a harmonic mapping in D and F = H + G ∈ B H . If h ′ + g ′ (cid:22) H ′ + G ′ , then we have || P f || ≤ || P F || . In this case, f is ULU in D . Proof.
By assumption, there exists a function φ ∈ A D such that h ′ = H ′ ◦ φ and g ′ = G ′ ◦ φ .Therefore, f is sense-preserving in D since F is sense-preserving in D . Moreover, we have P h = ( P H ◦ φ ) φ ′ and ω f ( z ) = g ′ ( z ) h ′ ( z ) = G ′ ( w ) H ′ ( w ) = ω F ( w ) , w = φ ( z ) . Consequently, ω f ( z ) ω ′ f ( z )1 − | ω f ( z ) | = ω F ( w ) ω ′ F ( w )1 − | ω F ( w ) | φ ′ ( z ) . It follows from (1.1) that P f = ( P F ◦ φ ) φ ′ . By Schwarz-Pick’s lemma, | φ ′ ( z ) | ≤ − | φ ( z ) | − | z | and using this, we find that(1 − | z | ) | P f ( z ) | = (1 − | z | ) | φ ′ ( z ) P F ( φ ( z )) | ≤ (1 − | φ ( z ) | ) | P F ( φ ( z )) | ≤ || P F || . The desired conclusion follows. (cid:3)
Often, the property of a sense-preserving harmonic mapping is mainly decided by itsanalytic part. As another example of it, we have
Theorem 9.2. (Subordination principle II)
Let f = h + g be a sense-preserving harmonicmapping in D and F = H + G ∈ B H such that h ′ (cid:22) H ′ . Then we have || P f || ≤ || P F || + 2 .Thus, f is ULU in D . Proof.
Since F ∈ B H , we know that H ∈ B A by Theorem 4.1. Clearly, B A ⊆ B H . Itfollows from the assumption and Theorem 9.1 that || P h || ≤ || P H || . Using the inequality(2.1) twice, we obtain that || P f || ≤ || P h || + 1 ≤ || P H || + 1 ≤ || P F || + 2and the proof is complete. (cid:3) Similar to Theorem 9.2, few other results on subordination of analytic functions in[24, Section 4] can be transplanted to the case of sense-preserving harmonic mappings byconsidering its analytic parts.
Acknowledgments.
The work was completed during the visit of the first author tothe Indian Statistical Institute, Chennai Centre and this author thanks the institute forthe support and the hospitality. The research of the first author was supported by theNSFs of China (No. 11571049), the Construct Program of the Key Discipline in HunanProvince, the Science and Technology Plan Project of Hunan Province (No. 2016TP1020)and the Science and Technology Plan Project of Hengyang City (2017KJ183). The workof the second author is supported by Mathematical Research Impact Centric Supportof DST, India (MTR/2017/000367). The second author is currently at Indian Statistical armonic mappings, the pre-Schwarzian and the Bloch space 27
Institute (ISI), Chennai Centre, Chennai, India. The authors thank the referee for his/hercomments.
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E-mail address : [email protected] S. Ponnusamy, Department of Mathematics, Indian Institute of Technology Madras,Chennai-600 036, India
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