Harmonic pre-Schwarzian and its applications
aa r X i v : . [ m a t h . C V ] J u l HARMONIC PRE-SCHWARZIAN AND ITS APPLICATIONS
GANG LIU AND SAMINATHAN PONNUSAMY
Abstract.
The primary aim of this article is to extend certain inequalities concerningthe pre-Schwarzian derivatives from the case of analytic univalent functions to that ofunivalent harmonic mappings defined on certain domains. This is done in two differentways. One of the ways is to connect with a conjecture on the univalent harmonic map-pings. Also, we improve certain known results on the majorization of the Jacobian offunctions in the affine and linear invariant family of sense-preserving harmonic mappings.This is achieved as an application of a corresponding distortion theorem in terms of theharmonic pre-Schwarzian derivative. Introduction
Throughout the paper, let D = { z : | z | < } , H = { z : Re z > } and ∆ = C \ D denoted the unit disk, the right half-plane and the exterior of the closed unit disk, respec-tively. Suppose f (resp. g ) is analytic in D (resp. H ) and F is analytic in ∆ \{∞} with asimple pole at z = ∞ . If f , g and F are univalent in D , H and ∆, respectively, then wehave sup z ∈ D (1 − | z | ) | f ′′ ( z ) /f ′ ( z ) | ≤ , (1.1)sup z ∈ H z | g ′′ ( z ) /g ′ ( z ) | ≤ , (1.2)and sup z ∈ ∆ ( | z | − | z | ) | F ′′ ( z ) /F ′ ( z ) | ≤ . (1.3)The constant 6 is sharp in all the three cases. Inequalities (1.1) and (1.2) are obtained asa consequence of Bieberbach’s distortion theorem. However, the estimate for F is deeperand is established in [2] as a consequence of Goluzin’s inequality [9, p. 139].Osgood [18] generalized (1.1) and (1.2) to an arbitrary simply connected domain bymeans of hyperbolic metric . In what follows, D ⊂ C is a domain with at least twoboundary points. By the uniformization theorem, the hyperbolic metric λ D ( z ) is inducedby λ D ( f ( z )) | f ′ ( z ) | = λ D ( z ) = 1 / (1 − | z | ) , z ∈ D , (1.4)where f : D → D is a (universal) covering mapping onto D . Especially, if D is simplyconnected, then f is a conformal mapping of D onto D . The definition is independent ofthe choice of the covering mapping of D onto D since two covering mappings of D differonly by a conformal self-mapping of D . Its Gaussian curvature is − Mathematics Subject Classification.
Primary: 31A05; Secondary 30C55.
Key words and phrases.
Pre-Schwarzian, harmonic mapping, hyperbolic metric, linear and affine in-variant, majorization, subordination.
Theorem A. ([18, Theorem 1])
Let f be a univalent analytic function in a simply con-nected domain D ⊂ C . Then we have the sharp inequality λ − D ( z ) | f ′′ ( z ) /f ′ ( z ) | ≤ , z ∈ D. (1.5)Furthermore, Osgood [18] stated that (1.5) does not hold for arbitrary domains. Forinstance, let D = D ∗ := D \{ } and f ( z ) = 1 /z . The basic theory of hyperbolic metricand computation shows that λ D ∗ ( z ) = 1 / (2 | z | log(1 / | z | )) and sup z ∈ D ∗ λ − D ∗ ( z ) | f ′′ ( z ) /f ′ ( z ) | = ∞ . Based on this, he investigated sufficient and necessary conditions to demonstrate thatsuch type of inequality does hold in a multiply connected domain.
Theorem B. ([18, Theorem 2])
Let D ⊂ C have at least two boundary points. Thereexists a constant a such that λ − D ( z ) | f ′′ ( z ) /f ′ ( z ) | ≤ a in D for all univalent analyticfunctions in D if and only if there exists a positive constant c such that λ D ( z ) d ( z, ∂D ) ≥ c, z ∈ D. (1.6) Here d ( z, ∂D ) denotes the Euclidean distance from z to the boundary ∂D of D . For the characterization of domains satisfying (1.6), see [18, Section 5]. The oper-ator f P f := f ′′ /f ′ , is called the pre-Schwarzian derivative of f when f is lo-cally univalent and analytic in D . The pre-Schwarzian norm of f in D is defined as || P f || D = sup z ∈ D λ − D ( z ) | P f ( z ) | . Recently, Hern´andez and Mart´ın [14] extended these no-tions to any locally univalent harmonic mapping in a domain (see also [7]). In view ofthis, in Section 3, we generalize those inequalities to complex-valued univalent harmonicmappings. Our results are based on a distortion theorem concerning the harmonic pre-Schwarzian derivative about linear and affine invariant family of harmonic mappings.However, the sharp results should be resorted to Clunie and Sheil-Small’s conjecture ([6])in terms specified order of the family S H of normalized univalent harmonic mappingsin the unit disk. In Section 4, we build some sharp inequalities on a class of analyticpre-Schwarzian derivatives and the harmonic pre-Schwarzian derivatives, and propose aconjecture about the relationship between univalent analytic functions and univalent har-monic mappings. We use these combinations to obtain similar results, as that of Section3, when the domain is simply connected. Furthermore, the proposed conjecture impliesthe former conjecture of Clunie and Sheil-Small. In Section 5, we improve the corre-sponding results of [22, Section 3] by applying our distortion theorem and the results on majorization-subordination theory of universal linear invariant family of analytic func-tions. 2. Background and Preliminaries
Univalent harmonic mappings.
A complex-valued function f is called a harmonicmapping if it satisfies the Laplace equation ∆ f = 4 f zz = 0. In a simply connected domain D , every harmonic mapping f has a decomposition f = h + g , where h and g are analyticfunctions in D . However, in a multiply connected domain, the representation f = h + g is valid locally but may not have a single-valued global extension. armonic pre-Schwarzian and its applications 3 According to Lewy’s theorem [15], a harmonic mapping f of the form f = h + g islocally univalent in a domain D if it is sense-preserving, i.e., its Jacobian J f = | h ′ | − | g ′ | is positive in D so that its dilatation ω f defined by ω f = g ′ /h ′ has the property that | ω f ( z ) | < D . In the study of univalent harmonic mappings, it is convenient toconsider the class S H of all sense-preserving and univalent harmonic mappings f = h + g in the unit disk D with the normalizations h (0) = h ′ (0) − g (0) = 0. The class S H isnot compact whereas S H := { f ∈ S H : g ′ (0) = 0 } is compact (see [8, p. 78]).Let K H (resp. C H ) be the set of all convex (resp. close-to-convex) harmonic mappingsfrom S H . Let K H := K H ∩ S H and C H := C H ∩ S H . Clunie and Sheil-Small [6] constructedthe harmonic half-plane mapping L and the harmonic Koebe function K defined by L ( z ) = 2 z − z − z ) + − z − z ) and K ( z ) = z − z + z (1 − z ) + z + z (1 − z ) , respectively. These functions play the role extremal in many problems of harmonic map-pings. Note that L ∈ K H and K ∈ C H . Basic information about harmonic mappings maybe obtained from the monograph of Duren [8] and the recent survey [20].2.2. Affine and linear invariant families.
Let F be a family of sense-preserving har-monic mappings f = h + g in D , normalized by h (0) = g (0) = h ′ (0) − F is said tobe a linear invariant family (LIF) if for each f ∈ F , K ϕ ( f ( z )) = f ( ϕ ( z )) − f ( ϕ (0)) ϕ ′ (0) h ′ ( ϕ (0)) ∈ F ∀ ϕ ∈ Aut( D ) , and F is called an affine invariant family (AIF) if for each f ∈ F , A ε ( f ( z )) = f ( z ) + εf ( z )1 + εg ′ (0) ∈ F ∀ ε ∈ D . Here K ϕ ( f ) and A ε ( f ) are called Koebe and affine transforms of f , respectively. We saythat F is an affine and linear invariant family (ALIF) if it is both LIF and AIF. Forexample, each of S H , K H and C H is an ALIF. The order of ALIF F , defined by α ( F ) = sup f ∈F | a ( f ) | = 12 sup f ∈F | h ′′ (0) | , plays an important role in the study of harmonic mappings, since the appearance of thepioneering work of Clunie and Sheil-Small (see [6]). In 2007, the notion of specified order of ALIF F was introduced in [10] as α ( F ) = sup f ∈F | a ( f ) | = 12 sup f ∈F | h ′′ (0) | , where F = { f = h + g ∈ F : g ′ (0) = 0 } . Note that 1 / ≤ α ( F ) ≤ α ( F ) ≤ α ( F ) + 1 / α ( F ) ≥ F (see [12]). Also the specified order α ( F ) of a given ALIF F coincides with the new order of LIF defined in [24]. Please refer to [11] for furtherdetails about α ( F ).It follows from [6, 25] that α ( K H ) = 3 / , α ( C H ) = 5 / , α ( K H ) = 2 and α ( C H ) = 3 . G. Liu and S. Ponnusamy
However, it is conjectured that α ( S H ) = 5 /
2, which is of special importance in obtainingsharp coefficient estimates for univalent harmonic mappings (see [6]). The upper boundfor α ( S H ) has been improved few times. See [6, p.10], [8, p. 96] and [23, Theorem 10].However, the conjectured bound remains open. Now the best known upper bound of itwas shown in [1].Let U α be the set of all locally univalent analytic functions h ( z ) = z + a z + · · · in D of order ≤ α , where α = sup h ∈U α | a ( h ) | = 12 sup h ∈U α | h ′′ (0) | . The family U α is known as the universal linear invariant family (ULIF) of order α ( ≥ h ∈ U α , then K ϕ ( h ( z )) ∈ U α holds for any ϕ ∈ Aut( D ). It is easy tosee that a ULIF is not an ALIF, but the set { h : f = h + g ∈ F } is a ULIF when F isan ALIF. Conversely, we can construct some special ALIFs from ULIFs. For instance, if F α := { f : f ( z ) = h ( z ) + b h ( z ) , h ∈ U α and b ∈ D } , (2.1)then it is easy to see that F α is an ALIF with α ( F α ) = α ( F α ) = α . Indeed, simplecomputation shows that K ϕ ( f ( z )) = K ϕ ( f ( z )) + B K ϕ ( f ( z )) ∈ F α , B = b ϕ ′ (0) h ′ ( ϕ ′ (0)) ϕ ′ (0) h ′ ( ϕ ′ (0)) , for each f ∈ F α and for any ϕ ∈ Aut( D ). This means that F α is a LIF. On the otherhand, calculations prove that F α is an AIF because A ε ( f ( z )) = h ( z ) + (cid:18) b + ε b ε (cid:19) h ( z ) ∈ F α for each f ∈ F α and for all ε ∈ D . Moreover, one can also check that α ( F α ) = α ( F α ) = α .2.3. Harmonic pre-Schwarzian derivatives.
Let f be a locally univalent harmonicmapping in a domain D . The harmonic pre-Schwarzian derivative of f and the harmonicpre-Schwarzian norm of f in D are defined by P f = (log J f ) z and || P f || D = sup z ∈ D λ − D ( z ) | P f ( z ) | , (2.2)respectively (see [14]). These definitions coincide with the corresponding definitions inthe analytic case.The harmonic pre-Schwarzian derivative inherits the same chain rule as in the analyticcase. More precisely, if f is a sense-preserving harmonic mapping and φ is a locallyunivalent analytic function for which the composition f ◦ φ is well defined, then, because J f ◦ φ = | φ ′ | J f ( φ ), we have P f ◦ φ = P f ( φ ) φ ′ + P φ . (2.3)The harmonic pre-Schwarzian derivative is invariant under an affine transformation ofharmonic mapping f : P A ◦ f = P f , A ( z ) = az + bz + c, | a | 6 = | b | . So does the harmonic pre-Schwarzian norm.Besides these, several of the recent properties on this topic may be found from [12, 14,16, 17] and from the references therein. In these articles, the authors focussed mainly on armonic pre-Schwarzian and its applications 5 sense-preserving harmonic mappings f = h + g in a simply connected domain D so thatthe first formulation in (2.2) can be rewritten as P f = P h − ωω ′ − | ω | , (2.4)where P h = h ′′ /h ′ and ω = ω f = g ′ /h ′ .3. Inequalities related to harmonic pre-Schwarzian derivatives
In this section, we will generalize results about (1.1), (1.2), Theorems A and B tounivalent harmonic mappings. However, we will give a negative answer to the case of(1.3) when F is a univalent harmonic mapping in ∆ with F ( ∞ ) = ∞ . The followinglemma may be considered as an adjustment of the result of Graf [12, Theorem 1] as canbe seen by a verification of the corresponding proof. So we omit the details. Lemma 1.
Let F be an ALIF and f ∈ F . Then | (1 − | z | ) P f ( z ) − z | ≤ α ( F ) , z ∈ D , (3.1) and || P f || D = sup z ∈ D (1 − | z | ) | P f ( z ) | ≤ α ( F ) + 1) . Both estimates are sharp if F = F α ( K H , C H ), where F α is defined by (2.1) . Recall that α ( F α ) = α . Let f α = k α + b k α , where b ∈ D and k α is defined by k α ( z ) = 12 α (cid:20) − (cid:18) − z z (cid:19) α (cid:21) , z ∈ D . (3.2)It is easy to see that f α ∈ F α and (3.1) is sharp for f α at z = 0. Moreover, we have || P f α || D = || P k α || D = 2( α + 1) . The sharpness part of (3.1) in the cases of K H and C H canbe obtained by setting z = 0 and by choosing the harmonic half-plane mapping L andthe harmonic Koebe mapping K , respectively. Note that P L ( z ) = 31 − z − z − | z | and P K ( z ) = 5 + 3 z − z − z − | z | . Simple computation and analysis show that || P L || D = 5 (see [14, Theorem 4]) and || P K || D = 7 (see [16, Theorem 1.1]). Since the harmonic pre-Schwarzian derivative pre-serves affine invariance and S H is a special ALIF, we can get the following result as acorollary to Lemma 1. This will be used in the sequel. Corollary 1.
Let f be a sense-preserving and univalent harmonic mapping in D . Then | (1 − | z | ) P f ( z ) − z | ≤ α ( S H ) , z ∈ D , (3.3) and || P f || D = sup z ∈ D (1 − | z | ) | P f ( z ) | ≤ α ( S H ) + 1) . (3.4)Clearly, (3.4) is a generalization of (1.1). Now, we will extend (1.2) to the case ofharmonic mappings. G. Liu and S. Ponnusamy
Theorem 1.
Let f be a sense-preserving and univalent harmonic mapping in the righthalf-plane H = { z : Re z > } . Then || P f || H = sup z ∈ H z | P f ( z ) | ≤ α ( S H ) + 1) . (3.5) Proof.
Fix z ∈ H . Then there exists a unique w ∈ D such that z = φ ( w ) = w − w . Thechain rule (2.3) and a basic computation show that2 Re z | P f ( z ) | = 2 Re( φ ( w )) | P f ( φ ( w )) | = (cid:18) w − w + 1 + w − w (cid:19) | φ ′ ( w ) | | P f ◦ φ ( w ) − P φ ( w ) | = (1 − | w | ) | P f ◦ φ ( w ) − P φ ( w ) | . Note that f ◦ φ is a sense-preserving and univalent harmonic mapping in D and P φ ( w ) =2 / (1 − w ). It follows from (3.3) that2 Re z | P f ( z ) | =(1 − | w | ) | P f ◦ φ ( w ) − P φ ( w ) |≤| (1 − | w | ) P f ◦ φ ( w ) − w | + | (1 − | w | ) P φ ( w ) − w |≤ α ( S H ) + 2 , which implies (3.5). (cid:3) However, there is no similar result to (1.3) when F is a univalent harmonic mapping in∆ = { z : | z | > } ∪ {∞} with F ( ∞ ) = ∞ . To demonstrate this fact, we consider F ( z ) = z − z + 2 log | z | , z ∈ ∆ . It follows from [13, Theorem 3.7] that F is a sense-preserving and univalent harmonicmapping in ∆. Direct computations reveal that J F ( z ) = | z | ( | z | − | z | and P F ( z ) = (log J F ( z )) z = 2 + z − | z | z ( z + 1)( | z | − . Obviously, sup z ∈ ∆ ( | z | − | z | ) | P F ( z ) | = ∞ . Next we consider the case of simply connected domain.
Theorem 2.
Let f be a sense-preserving and univalent harmonic mapping in a simplyconnected domain D ⊂ C . Then λ − D ( z ) | P f ( z ) | ≤ α ( S H ) + 2) , z ∈ D. (3.6) Proof.
Fix z ∈ D and choose a conformal mapping φ of D onto D with φ (0) = z .Distortion theorem of Bieberbach for the univalent function φ gives that (cid:12)(cid:12)(cid:12)(cid:12) (1 − | ζ | ) φ ′′ ( ζ ) φ ′ ( ζ ) − ζ (cid:12)(cid:12)(cid:12)(cid:12) ≤ , ζ ∈ D , armonic pre-Schwarzian and its applications 7 which implies | P φ (0) | = | φ ′′ (0) /φ ′ (0) | ≤
4. Moreover, by (1.4), we have λ − D ( z ) = | φ ′ (0) | .Note that f ◦ φ is a sense-preserving and univalent harmonic mapping in D , and thus,(3.3) holds for f ◦ φ . In particular, (3.3) applied to f ◦ φ at the point z = 0 yields | P f ◦ φ (0) | ≤ α ( S H ) . By (2.3), we have λ − D ( z ) | P f ( z ) | = | φ ′ (0) P f ( φ (0)) | = | P f ◦ φ (0) − P φ (0) |≤ | P f ◦ φ (0) | + | P φ (0) |≤ α ( S H ) + 4and the desired inequality (3.6) follows. (cid:3) As remarked in the introduction about Theorem B, Theorem 2 cannot be extendedto arbitrary domains. However, using the method of proof of Theorem B (we omit thedetail), we obtain a similar result (see Theorem 3 below) based on the following lemma,which is a generalization of [18, Lemma 1].
Lemma 2. If D is a proper subdomain of C and if f is a sense-preserving and univalentharmonic mapping in D , then | P f ( z ) | ≤ α ( S H ) d ( z, ∂D ) , z ∈ D, (3.7) where d ( z, ∂D ) denotes the Euclidean distance to the boundary. Proof.
Fix z ∈ D and d = d ( z , ∂D ). Then F ( z ) = f ( d z + z ) is a sense-preservingand univalent harmonic mapping in D . Using (2.3) and (3.3), we get | P f ( z ) | = | P F (0) | d ≤ α ( S H ) d = 2 α ( S H ) d ( z, ∂D )and the proof is complete. (cid:3) Theorem 3.
Let D ⊂ C have at least two boundary points. Then there exists a constant a such that λ − D ( z ) | P f ( z ) | ≤ a in D for all sense-preserving and univalent harmonicmappings in D if and only if there exists a positive constant c such that λ D ( z ) d ( z, ∂D ) ≥ c, z ∈ D. Remarks.
The inequalities (3.3)-(3.7) are concerned with the conjecture on α ( S H ). Forexample, if α ( S H ) = 5 /
2, then these inequalities are sharp. In what follows, K denotesthe harmonic Koebe function.(1) The inequality (3.3) is sharp for the function K (Set z = 0 in (3.3)).(2) The sharpness of (3.4) can be easily seen from the function K .(3) The sharpness in (3.5) follows by choosing f = K ◦ φ with φ ( z ) = (1 − z ) / (1 + z ).In order to verify this, we may use (2.2) and observe that J f ( z ) = z + z | z | and P f ( z ) = − z + 4 zz ( z + z ) . G. Liu and S. Ponnusamy (4) To show the sharpness of (3.6), we choose D = C \ ( −∞ ,
0] and consider thefunction f ( z ) = ( K ◦ φ ◦ φ )( z ) = 124 (cid:18) z / + 3 z − (cid:19) + 124 (cid:18) z / − z + 1 (cid:19) , z ∈ D, where φ ( z ) = √ z (arg 1 = 0) and φ ( z ) = − z z . Note that φ (resp. φ ) is aconformal mapping of D (resp. H ) onto H (resp. D ). Since K is univalent in D , f is univalent in D . By (2.2), straightforward computations assert that J f ( z ) = √ z + √ z | z | and P f ( z ) = − | z | + 4 z z ( | z | + z ) , z ∈ D. Note that λ − D ( z ) = 4 z and λ − D ( x ) | P f ( x ) | = 9 for all x > D = D and f = K show that (3.7) is also sharp.4. Inequalities between analytic pre-Schwarzian and harmonicpre-Schwarzian
In this section, we try to find some connections between univalent analytic functionsand univalent harmonic mappings. For this, we first investigate certain relationshipsbetween the class of analytic pre-Schwarzian derivatives and the harmonic pre-Schwarzianderivative of a given sense-preserving harmonic mapping.
Theorem 4.
Let f = h + g be a sense-preserving harmonic mapping in a simply connecteddomain D ⊂ C with the dilatation ω f . Then for each ε ∈ D we have λ − D ( z ) | P h + εg ( z ) − P f ( z ) | ≤ sup z ∈ D | ω f ( z ) | , z ∈ D. (4.1) Moreover, either || P h + εg || D = || P f || D = ∞ or both || P h + εg || D and || P f || D are finite. If || P f || D < ∞ , then the inequality (cid:12)(cid:12) || P h + εg || D − || P f || D (cid:12)(cid:12) ≤ sup z ∈ D | ω f ( z ) | (4.2) holds. Furthermore, for any given k ∈ [0 , , there exists an ε k ∈ D and a sense-preservingharmonic mapping F k = H k + G k in D with sup z ∈ D | ω F k ( z ) | = k such that (4.1) is sharp,and (4.2) is sharp when D is a disk. Proof.
Let f = h + g be sense-preserving in D . Then its dilatation ω := ω f : D → D isanalytic and k = sup z ∈ D | ω ( z ) | exists, where k ∈ [0 , k = 0 andthus, we assume that k ∈ (0 , ε ∈ D , we observe that | h ′ ( z ) + εg ′ ( z ) | ≥ | h ′ ( z ) | − | g ′ ( z ) | > , z ∈ D, so that h + εg is locally univalent in D . Fix ε ∈ D . From (2.4), direct computation showsthat 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 | . armonic pre-Schwarzian and its applications 9 We first consider the special case D = D . Clearly, sup z ∈ D (cid:12)(cid:12)(cid:12) ε + ω ( z )1+ εω ( z ) (cid:12)(cid:12)(cid:12) ≤
1. On the otherhand, applying Schwarz-Pick lemma to the function ω/k : D → D , we infer that | ω ′ ( z ) | (1 − | z | )1 − | ω ( z ) | ≤ k − | ω ( z ) k | − | ω ( z ) | ≤ k, z ∈ D . This means that (4.1) holds for D = D . Using the triangle inequality, it follows that(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 ) | ≤ k, z ∈ D . By (2.2), it is easy to see that either || P h + εg || D = || P f || D = ∞ or both || P h + εg || D and || P f || D are finite. Moreover, if || P f || D < ∞ , then (4.2) can be deduced from the aboveinequality when D = D .Next we will discuss the sharpness part. For any given k ∈ (0 , f k defined on D by f k ( z ) = h k ( z ) + g k ( z ) = Z z (1 + kt ) a (1 − kt ) a +1 dt + k Z z t (1 + kt ) a (1 − kt ) a +1 dt, (4.3)where a ≥
0. Obviously, ω f k ( z ) = kz and thus, f k is sense-preserving in D . By computa-tions, we have P h k + g k ( z ) = 2 k ( a + 1)1 − k z and P f k ( z ) = k a + 1 + kz − k z − k z − k | z | so that (cid:2) (1 − | z | ) (cid:12)(cid:12) P h k + g k ( z ) − P f k ( z ) | (cid:3) z =0 = k, which shows that (4.1) is sharp when D = D . On the other hand, since(1 − | z | ) | P h k + g k ( z ) | ≤ (1 − | z | ) 2 k ( a + 1)1 − k | z | ≤ k ( a + 1) , z ∈ D , and (cid:2) (1 − | z | ) | P h k + g k ( z ) | (cid:3) z =0 = 2( a + 1) k, we see that || P h k + g k || D = 2( a + 1) k . Similarly, we obtain || P h k − g k || D = 2 ak . It follows from(4.2) that || P h k + g k || D − k ≤ || P f k || D ≤ || P h k − g k || D + k, which means that || P h k + g k || D − k = || P f k || D = (2 a + 1) k = || P h k − g k || D + k. This certifies the sharpness of (4.2) when D = D .Now we need to consider the general case. Fix z ∈ D and consider a conformal mapping φ of D onto D with φ (0) = z . For simplicity, let f ε = h + εg . Applying (1.4) and (2.3),we have λ − D ( z ) | P f ε ( z ) − P f ( z ) | = λ − D ( z ) | P ( f ε ◦ φ ) ◦ ψ ( z ) − P ( f ◦ φ ) ◦ ψ ( z ) | = | ψ ′ ( z ) | λ − D ( z ) | P f ε ◦ φ ( ψ ( z )) − P f ◦ φ ( ψ ( z )) | = λ − D ( ψ ( z )) | P f ε ◦ φ ( ψ ( z )) − P f ◦ φ ( ψ ( z )) | , where ψ = φ − : D → D is the inverse function of φ . Thus, (4.1) follows easily because f ◦ φ is a sense-preserving harmonic mapping in D . To show the sharpness of (4.1), itsuffices to consider the function F k = f k ◦ ψ , where f k is defined by (4.3).If D is a disk, then, without loss of generality, we may assume D = { z : | z − z |
Corollary 2.
Let f = h + g be a sense-preserving harmonic mapping in a simply connecteddomain D ⊂ C with the dilatation ω f . Then for each pair ε , ε ∈ D , we have λ − D ( z ) (cid:12)(cid:12) P h + ε g ( z ) − P h + ε g ( z ) (cid:12)(cid:12) ≤ z ∈ D | ω f ( z ) | , z ∈ D. (4.4) Moreover, either || P h + ε g || D = || P h + ε g || D = ∞ or both || P h + ε g || D and || P h + ε g || D arefinite. If || P h + εg || D < ∞ for some ε ∈ D , then (cid:12)(cid:12) || P h + ε g || D − || P h + ε g || D (cid:12)(cid:12) ≤ z ∈ D | ω f ( z ) | (4.5) holds for any ε , ε ∈ D . Furthermore, for any given k ∈ [0 , , there exist ε ( k ) , ε ( k ) ∈ D and a sense-preserving harmonic mapping F k = H k + G k in D with sup z ∈ D | ω F k ( z ) | = k such that (4.4) is sharp, and (4.5) is sharp when D is a disk. Corollary 3.
For any sense-preserving harmonic mapping f = h + g in a simply connecteddomain D ⊂ C , we have λ − D ( z ) | P h + εg ( z ) − P f ( z ) | ≤ ∀ ε ∈ D (4.6) in D and max (cid:26) , max ε ∈ D || P h + εg || D − (cid:27) ≤ || P f || D ≤ min ε ∈ D || P h + εg || D + 1 . (4.7) The constant 1 is sharp in the two estimates.
In particular, if f = h + g in Corollary 3 is further restricted to be univalent, then, forany given ε ∈ D , the distance between || P f || D and || P h + εg || D is at most 1. Compared tothe corresponding results in Sections 1 and 3, if the conjecture of Clunie and Shell-Small[6] on α ( S H ) were true, then the distance between the sharp constant 6 (resp. 8) in (1.1) armonic pre-Schwarzian and its applications 11 and (1.2) (resp. (1.5)) and the sharp constant 2( α ( S H ) + 1) (resp. 2( α ( S H ) + 2)) in(3.4) and (3.5) (resp. (3.6)) is also 1. This raises the following conjecture. Conjecture 1.
Let f = h + g be a sense-preserving and univalent harmonic mapping ina simply connected domain D ⊂ C . Then there exists a constant ε ∈ D such that h + εg is univalent in D . It is easy to see that to solve the above conjecture, it suffices to consider the case D = D .Moreover, this conjecture is weaker than the following conjecture proposed in [21]. Conjecture A.
For every function f = h + g ∈ S H , there exists a constant θ ∈ R suchthat h + e iθ g ∈ S , where S is the class of analytic functions in S H . If Conjecture 1 were true, then, combining (4.6) with Bieberbach’s distortion theoremfor univalent analytic functions in D , we can obtain the sharp inequality (3.3). Moreover,combining (4.7) (resp. (4.6)) with (1.1) and (1.2) (resp. (1.5)), we can obtain sharpinequalities (3.4) and (3.5) (resp. (3.6)), respectively. In fact, Conjecture 1 implies α ( S H ) = 5 /
2. To clarify this, let us assume for the moment that Conjecture 1 is true.Then, for f = h + g ∈ S H , there exists an ε ∈ D such that h + ε g is univalent in D . Let D = D in (4.6). Then we have that | P h + ε g (0) − P f (0) | ≤
1. Using the method of proofof Theorem 2, we can obtain that | P h + ε g (0) | ≤
4. Note that | P f (0) | = | P h (0) | = 2 | a ( f ) | ,because of ω f (0) = 0. From this observation, it is easy to see that | a ( f ) | ≤ / K belongs to S H with a ( K ) = 5 / α ( S H ) = 5 / S H (see [6]). 5. Applications
In this section, we improve the corresponding results of [22, section 3] where the authorused the ALIF of harmonic mappings to obtain the radius of majorization of the Jacobianof harmonic mappings. A function f is said to be majorized by F in a certain region if | f ( z ) | ≤ | F ( z ) | holds there. Next let’s recall the notion of subordination. Let f and F be two harmonic mappings in D . We say that f is subordinate to F , denoted by f ≺ F ,if f ( z ) = F ( ψ ( z )), where ψ is analytic with ψ (0) = 0 and | ψ ( z ) | < D . For theimportance, background, development and results concerning these two topics, the readermay refer to the paper [22] and the references therein.Below, we denote n ( x ) = 1 + x − √ x + 2 x , x ≥
0. One of the subordination-majorization results for ULIF of analytic functions is the following.
Theorem C.
Let U α be a ULIF with ≤ α ( U α ) < ∞ . If f ≺ F and F ∈ U α , then | f ′ ( z ) | ≤ | F ′ ( z ) | for | z | ≤ n ( α ) , α = α ( U α ) and the result is best possible. The case α ( U α ) ≥ .
65 in Theorem C was established first by Campbell (see [5]).Also, he conjectured that Theorem C held for 1 ≤ α ( U α ) < .
65, which was later provedaffirmatively by Barnard and Pearce (see [4]). By the way, the case α ( U α ) = 1 wasdealt in [3]. It is natural to ask for the analogous result to ALIF of harmonic mappings.Schaubroeck [22] has obtained a partial answer. We will improve his results based on the following theorem, which is exactly the same as [11, Theorem 1], but we present a muchsimpler proof of it. Theorem 5.
Let F be an ALIF and f ∈ F . Then (1 − r ) α − (1 + r ) α +2 ≤ J f ( z )1 − | b | ≤ (1 + r ) α − (1 − r ) α +2 , | z | = r < , (5.1) where α = α ( F ) and b = f z (0) . Equalities occur if F = F α for the functions f ( z ) = k α ( z ) + b k α ( z ) , where F α and k α are defined by (2.1) and (3.2) , respectively. In addition,equalities hold if F = K H (resp. C H ) for the functions f ( z ) = L ( z ) + b L ( z ) (cid:16) resp. K ( z ) + b K ( z ) (cid:17) , where L and K are the harmonic half-plane mapping and the harmonic Koebe mapping,respectively. Proof.
Let z = re iθ ∈ D . It follows from (3.1) that | z (1 − | z | ) P f ( z ) − | z | | ≤ α | z | andthus 2 r − αr − r ≤ Re zP f ( z ) ≤ αr + 2 r − r , | z | = r < . (5.2)Note that 12 r ∂∂r log J f ( z ) = r P f ( z ) e iθ + P f ( z ) e − iθ ) = Re zP f ( z ) . If we substitute the above equality into (5.2) and integrate the resulting inequalities, weobtain the desired conclusion. (cid:3)
The following result is a generalization of Theorem C. The proof of this result followsif we adopt the method of the proof of [22, Theorem 3.4] carefully. For the sake ofcompleteness, we include the details here.
Theorem 6.
Let F be an ALIF with ≤ α ( F ) < ∞ . If f ≺ F and F ∈ F , then J f ( z ) ≤ J F ( z ) for | z | ≤ n ( α ( F )) and the result is best possible if F = F α , where F α isdefined by (2.1) . Proof.
Fix F = H + G ∈ F and consider S ( F )( z ) = A ε ◦ K ϕ ( F ( z )) , ϕ ( z ) = z + z z z , ε = − G ′ ( ϕ (0)) H ′ ( ϕ (0)) , where | z | < K ϕ and A ε are defined as in Section 2.2. It is easy to see that S ( F ) ∈ F .Denote α ( F ) by α . It follows from (5.1) that J S ( F ) ( z ) = | ϕ ′ ( z ) | (1 − | z | ) · J F ( ϕ ( z )) J F ( z ) ≤ (1 + | z | ) α − (1 − | z | ) α +2 . (5.3)If we substitute z = ( x − z ) / (1 − z x ) into (5.3), then a direct calculation shows that J F ( x ) J F ( z ) ≤ (cid:18) − | z | − | x | (cid:19) (cid:18) | ( x − z ) / (1 − z x ) | − | ( x − z ) / (1 − z x ) | (cid:19) α . (5.4) armonic pre-Schwarzian and its applications 13 If f ≺ F , then f ( z ) = F ( ψ ( z )) and thus, J f ( z ) = | ψ ′ ( z ) | J F ( ψ ( z )). Applying (5.4) at x = ψ ( z ), we obtain that s J f ( z ) J F ( z ) ≤ | ψ ′ ( z ) | − | z | − | ψ ( z ) | (cid:18) | − z ψ ( z ) | + | ψ ( z ) − z || − z ψ ( z ) | − | ψ ( z ) − z | (cid:19) α . The quantity on the right side above is not greater than 1 for | z | ≤ n ( α ) (see [4] and [5]).To show that the result is best possible for F = F α , it suffices to consider the followingfunctions F ( z ) = k α ( z ) + b k α ( z ) and f a ( z ) = F ( ψ ( z )) := h a ( z ) + b h a ( z ) , where b ∈ D , k α is defined by (2.2) and ψ ( z ) = z ( a + z ) / (1 + az ), 0 ≤ a ≤
1. Clearly, f a ( z ) ≺ F ( z ) for any a ∈ [0 ,
1] and F ∈ F α . Note that h a ( z ) = k α ( ψ ( z )) , J F ( z ) = (1 − | b | ) | k ′ α ( z ) | and J f a ( z ) = (1 − | b | ) | h ′ a ( z ) | . It follows from the proof of [5, p.302-303] that the result in Theorem C is best possible.Therefore, the result in Theorem 6 is also best possible if F = F α . (cid:3) Corollary 4. If f ≺ F and F ∈ K H (resp. C H ), then J f ( z ) ≤ J F ( z ) for | z | ≤ n (3 /
2) =(5 − √ / ≈ . (resp. n (5 /
2) = (7 − √ / ≈ . ) and the result is bestpossible. Proof.
Note that α ( K H ) = 3 / α ( C H ) = 5 / K H cannot be improved. For this, we let F ( z ) = − L ( − z ) and f a ( z ) = F ( ψ ( z )), where L is the harmonic half-plane mapping and ψ ( z ) = z ( a + z ) / (1 + az ), 0 ≤ a ≤
1. Obviously, f a ≺ F for each a ∈ [0 ,
1] and F ∈ K H .Direct computations give for 0 < r < J f a ( r ) = (1 − r ) ( a + 2 r + ar ) (1 + 2 ar + r ) and (cid:20) ∂∂a J f a ( r ) (cid:21) a =1 = 2 1 − r (1 + r ) (1 − r + r )which is negative in the interval (0 ,
1) if n (3 / < r <
1. This means that J f a ( r ) is adecreasing function of a for r in this interval. Therefore, for any given sufficient small ε >
0, there exists a constant a ε which is sufficiently close to 1 such that J f aε ( r ) > J f ( r ) = 1 − r (1 + r ) = J F ( r ) , n (3 /
2) + ε < r < . Thus, J f aε is not majorized by J F in n (3 /
2) + ε < | z | < n (3 /
2) is best possible.Let us next indicate the sharpness of the result for the family C H . Indeed, similaranalysis for the family C H may be considered by setting F ( z ) = − K ( − z ) and f a ( z ) = F ( ψ ( z )), where K is the harmonic Koebe function, ψ ( z ) = z ( a + z ) / (1 + az ) with 0 ≤ a ≤
1, and for 0 < r < J f a ( r ) = (1 − r ) ( a + 2 r + ar ) (1 + 2 ar + r ) and (cid:20) ∂∂a J f a ( r ) (cid:21) a =1 = 2 (1 − r ) (1 + r ) (1 − r + r ) . This completes the proof. (cid:3)
Acknowledgments.
The work was completed during the visit of the first author to theIndian Statistical Institute, Chennai Centre and this author thanks the institute for thesupport and the hospitality. The research of the first author was supported by the NSFsof China (No. 11571049), the Construct Program of the Key Discipline in Hunan Provinceand the Science and Technology Plan Project of Hunan Province (No. 2016TP1020). Thesecond author is on leave from IIT Madras, Chennai.
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G. Liu, College of Mathematics and Statistics (Hunan Provincial Key Laboratoryof Intelligent Information Processing and Application), Hengyang Normal University,Hengyang, Hunan 421002, China.
E-mail address : [email protected] S. Ponnusamy, Stat-Math Unit, Indian Statistical Institute (ISI), Chennai Centre, 110,Nelson Manickam Road, Aminjikarai, Chennai, 600 029, India.
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