Boundary Schwarz lemma for harmonic mappings having zero of order p
aa r X i v : . [ m a t h . C V ] J u l BOUNDARY SCHWARZ LEMMA FOR HARMONIC MAPPINGSHAVING ZERO OF ORDER p XIAO-JIN BAI, JIE HUANG, AND JIAN-FENG ZHU
Abstract.
Suppose w is a sense-preserving harmonic mapping of the unit disk D such that w ( D ) ⊆ D and w has a zero of order p ≥ z = 0. In this paper,we first improve the Schwarz lemma for w , and then, we establish its boundarySchwarz lemma. Moreover, by using the automorphism of D , we further generalizethis result. Introduction
Let D = { z ∈ C : | z | < } be the unit disk, T = { z ∈ C : | z | = 1 } the unitcircle, and D the closure of D , i.e., D = D ∪ T . For z ∈ D , the formal derivatives ofa complex-valued function f are defined by: f z = 12 (cid:18) ∂f∂x − i ∂f∂y (cid:19) and f ¯ z = 12 (cid:18) ∂f∂x + i ∂f∂y (cid:19) . For each α ∈ [0 , π ], the directional derivative of f at z ∈ D is defined by ∂ α f ( z ) = lim r → + f ( z + re iα ) − f ( z ) r = e iα f z ( z ) + e − iα f ¯ z ( z ) . Then max ≤ α ≤ π {| ∂ α f ( z ) |} = Λ f ( z ) = | f z ( z ) | + | f ¯ z ( z ) | and min ≤ α ≤ π {| ∂ α f ( z ) |} = λ f ( z ) = (cid:12)(cid:12) | f z ( z ) | − | f ¯ z ( z ) | (cid:12)(cid:12) . A function f is said to be locally univalent and sense-preserving in D if and only ifits Jacobian J f satisfies the following condition (cf. [8]): For any z ∈ D , J f ( z ) = | f z ( z ) | − | f ¯ z ( z ) | > . Here and hereafter, the notation C m ( E ) denotes the set of all functions which are m -times continuously differentiable in domain E ⊂ C , where m ≥ C ( E ), which is always denoted by C ( E ), means the set of all continuousfunctions in E . Mathematics Subject Classification.
Primary: 30C15; Secondary: 31A20, 30C62.
Key words and phrases.
Schwarz lemma, boundary Schwarz lemma, harmonic mappings, mul-tiplicity of zeros.File: BHZ-20-0724.tex, printed: 28-7-2020, 1.18.
A function w ∈ C ( E ) is said to be harmonic in E if it satisfies the followingLaplace equation ∆ w = 4 w z ¯ z = 0 . Obviously, harmonic mappings are generalizations of analytic functions.In a simply connected domain Ω ⊂ C , a harmonic mapping w has the represen-tation w = h + ¯ g , where h and g are analytic in Ω. Furthermore, if g (0) = 0, thenthe representation is unique and called the canonical representation . We refer to [5]for more properties of harmonic mappings.In the rest of this paper, we use w to stand for the harmonic mappings of D , and f to stand for the analytic function of D .1.1. The multiplicity of zeros for analytic functions and harmonic map-pings.
Analytic case.
Suppose that f is an analytic function of D . Then f is saidto have a zero of order n at z , where n ≥
1, denoted by µ ( z , f ) = n , if f ( z ) = Df ( z ) = · · · = D n − f ( z ) = 0 and D n f ( z ) = 0, i.e., f ( z ) = ∞ X k = n a k ( z − z ) k , for z ∈ D . Here and hereafter the symbol D k f (resp. ¯ D k f ) means the k − th order derivativewith respect to z (resp. ¯ z ) of the complex-valued function f , i.e., D k f = ( ∂∂z ) k f ( z )(resp. ¯ D k f = ( ∂∂ ¯ z ) k f ( z )).The following result is a consequence of the Schwarz-Pick lemma applied to thefunction f /z p (cf. [6, Corollary 1.3] or [12, Remark 3]). Lemma A.
Let f : D → D be an analytic function with µ (0 , f ) = p ≥ . Then forany z ∈ D , | f ( z ) | ≤ | z | p | z | + | a p | | a p || z | , where a p = D p f (0) p ! . Harmonic case.
Suppose that w = h + ¯ g is a harmonic mapping of D . Forany z ∈ D , let ω ( z ) = w ¯ z ( z ) w z ( z )be the second complex dilatation of w . Then ω ( z ) = g ′ ( z ) h ′ ( z ) is an analytic function in D . Moreover, if w ( z ) is sense-preserving, then | ω ( z ) | < z ∈ D .We now introduce the definition of the multiplicity for sense-preserving harmonicmappings w in D . Suppose that w = h + ¯ g is a sense-preserving harmonic mappingof D , where h and g have respectively multiplicity n and m at z with w ( z ) = 0,i.e., h ( z ) = ∞ X k = n a k ( z − z ) k , g ( z ) = ∞ X k = m b k ( z − z ) k , z ∈ D . oundary Schwarz lemma for harmonic mappings having zero of order p Then n < m or m = n and | b n | < | a n | , since | ω ( z ) | <
1. We say that w has a zeroof order n at z and write µ ( z , w ) = n .The following lemma is due to Ponnusamy and Rasila [13]. Note that if p = 1,then it is the well-known harmonic version of the classical Schwarz lemma due toHeinz [7]. Lemma B.
Let w be a sense-preserving harmonic mapping of D such that µ (0 , w ) = p ≥ and w ( D ) ⊂ D . Then for any z ∈ D , | w ( z ) | ≤ π arctan | z | p ≤ π | z | p . Using Lemma A, we first improve Lemma B as follows:
Lemma 1.1.
Let w = h + ¯ g be a sense-preserving harmonic mapping of D such that µ (0 , w ) = p ≥ and w ( D ) ⊂ D . Then for any z ∈ D , (1.1) | w ( z ) | ≤ π arctan (cid:20) | z | p | z | + π ( | a p | + | b p | )1 + π ( | a p | + | b p | ) | z | (cid:21) , where a p = D p h (0) p ! and b p = D p g (0) p ! . Since w is a harmonic self-mapping of D , it follows from [2, Lemma 1] that(1.2) | a n | + | b n | ≤ π , for all n = 1 , , · · · . For any 0 ≤ r <
1, the function ϕ ( x ) = r + π x π xr is an increasing function of x , thenwe see that 4 π arctan (cid:20) | z | p | z | + π ( | a p | + | b p | )1 + π ( | a p | + | b p | ) | z | (cid:21) ≤ π arctan | z | p . The boundary Schwarz lemma for analytic functions and harmonicmappings.
Let us recall the following classical boundary Schwarz lemma for ana-lytic functions, which was proved in [6].
Theorem C. ([6, Page 42])
Suppose f : D → D is an analytic function with f (0) =0 , and, further, f is analytic at z = 1 with f (1) = 1 . Then, the following twoconclusions hold:(1) f ′ (1) ≥ .(2) f ′ (1) = 1 if and only if f ( z ) ≡ z . Theorem C has the following generalization.
Theorem D. ([9, Theorem 1.1 ′ ]) Suppose f : D → D is an analytic function with f (0) = 0 , and, further, f is analytic at z = α ∈ T with f ( α ) = β ∈ T . Then, thefollowing two conclusions hold:(1) βf ′ ( α ) α ≥ .(2) βf ′ ( α ) α = 1 if and only if f ( z ) ≡ e iθ z , where e iθ = βα − and θ ∈ R . Bai, Huang and Zhu
We remark that, when α = β = 1, Theorem D coincides with Theorem C.This useful result has attracted much attention and has been generalized in variousforms (see, e.g., [1, 3, 4, 10, 11, 16]). Recently, Wang et. al. obtained the boundarySchwarz lemma for solutions to the Poisson’s equation ([15]). By analogy with thestudies in the above results, in this paper, we discuss the boundary Schwarz lemmafor harmonic mappings having a zero of order p . Our main results are as follows: Theorem 1.1.
Let w = h + ¯ g be a sense-preserving harmonic mapping of D suchthat µ (0 , w ) = p ≥ and w ( D ) ⊂ D . If w is differentiable at z = 1 with w (1) = 1 ,then Re [ w z (1) + w ¯ z (1)] ≥ π ( p + 1) + π ( p − | a p | + | b p | )1 + π ( | a p | + | b p | ) , where a p = D p h (0) p ! and b p = D p g (0) p ! . For p = 1, it follows from (1.2) that | a | + | b | ≤ π . By using Theorem 1.1, wethen have Re [ w z (1) + w ¯ z (1)] ≥ π
21 + π ( | a | + | b | ) ≥ π . Theorem 1.2.
Let w = h + ¯ g be a sense-preserving harmonic mapping of D suchthat µ ( a, w ) = p ≥ and w ( D ) ⊂ D , where a ∈ D . If w is differentiable at z = α with w ( α ) = β , where α, β ∈ T , then Re (cid:18) ¯ β [ w z ( α ) α + w ¯ z ( α ) ¯ α ] (cid:19) ≥ π ( p + 1) + π ( p − ( p ) w ( a )(1 − | a | ) p π Λ ( p ) w ( a )(1 − | a | ) p − | a | | − ¯ aα | , where Λ ( p ) w ( a ) = (cid:12)(cid:12)(cid:12) D p h ( a ) p ! (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) D p g ( a ) p ! (cid:12)(cid:12)(cid:12) .In particular, when α = β = 1 and a = 0 , then Theorem . coincides withTheorem . . The rest of this paper is organized as follows: in Section 2 we shall introduce someknown results and prove two lemmas which will be used in the proof of our mainresults; in Section 3 we should prove Lemma 1.1, Theorem 1.1 and Theorem 1.2.2.
Auxiliary results
The following lemmas will be used in proving our main results.
Lemma 2.1. [14, Theorem 2] If m ( t ) and q ( t ) are functions for which all the nec-essary derivatives are defined, then D n m ( q ( t )) = X k + ··· + nk n = n n ! k ! · · · k n ! ( D k + ··· + k n m )( q ( t )) (cid:18) D ( q ( t ))1! (cid:19) k · · · (cid:18) D n ( q ( t )) n ! (cid:19) k n , where k , · · · , k n are non-negative integer numbers. Lemma 2.2.
Let S = { w ∈ C : | Re( w ) | < } be a strip domain, and f : D → S bean analytic function such that µ (0 , f ) = p ≥ . Assume that δ ( z ) = tan (cid:0) π f ( z ) (cid:1) .Then δ ( z ) is analytic in D with δ ( D ) ⊂ D and µ (0 , δ ) = p ≥ . oundary Schwarz lemma for harmonic mappings having zero of order p Proof.
We first prove that δ ( z ) is analytic in D and δ ( D ) ⊂ D .To show this, assume that f ( z ) = u + iv and let ζ = e πf i = e − πv e πu i . Since f is an analytic function of D into S , we see that ζ is an analytic function of D into H + = { ζ ∈ C : Re ζ > } . This implies that δ ( z ) is analytic in D , since δ ( z ) = ( − i ) ζ ( z ) − ζ ( z ) + 1 . The M¨obius transformation ζ − ζ +1 maps H + into D , and thus, δ ( D ) ⊂ D .Secondly, we show that µ (0 , δ ) = p .Obviously, δ (0) = 0. Let ϕ ( z ) = π f ( z ). It follows from Lemma 2.1 that D n δ = X k + ··· + nk n = n n ! k ! · · · k n ! (cid:0) D k + ··· + k n tan (cid:1) ( ϕ ) (cid:18) Dϕ (cid:19) k · · · (cid:18) D n ϕn ! (cid:19) k n = X k + ··· + nk n = nk n =0 n ! k ! · · · k n ! (cid:0) D k + ··· + k n tan (cid:1) ( ϕ ) (cid:18) Dϕ (cid:19) k · · · (cid:18) D n ϕn ! (cid:19) k n + ( D tan)( ϕ ) D n ϕ. The condition µ (0 , f ) = p ensures that D k ϕ (0) = 0, for k = 1 , , · · · , p − D p ϕ (0) = 0. Therefore, D n δ (0) = 0, for n = 1 , , · · · , p −
1. For n = p , we have(2.1) D p δ (0) = ( D tan)( ϕ (0)) D p ϕ (0) = D p ϕ (0) = 0 , which shows that µ (0 , δ ) = p . (cid:3) Given a ∈ D , let η ( z ) = ϕ a ( z ) = a − z − ¯ az be an automorphism of D , which inter-changes a and z . Then we have the following lemma. Lemma 2.3.
Let w = h + ¯ g be a sense-preserving harmonic mapping of D such that µ ( a, w ) = p ≥ and w ( D ) ⊂ D , where a ∈ D . Assume that W = w ◦ η . Then W isa sense-preserving harmonic self-mapping of D and µ (0 , W ) = p. Proof.
Obviously, W (0) = w ( η (0)) = w ( a ) = 0 . Elementary calculations show that, | W z ( z ) | − | W ¯ z ( z ) | = ( | w η ( η ) | − | w ¯ η ( η ) | ) | ϕ ′ a ( z ) | . Bai, Huang and Zhu
Since w is sense-preserving in D , we see that | W z | − | W ¯ z | >
0, and thus, W is alsosense-preserving in D . Using Lemma 2.1, we obtain D n W = X k + ··· + nk n = n n ! k ! · · · k n ! ( D k + ··· + k n w )( η ) (cid:18) Dη (cid:19) k · · · (cid:18) D n ηn ! (cid:19) k n = X k + ··· + nk n = nk = n n ! k ! · · · k n ! ( D k + ··· + k n w )( η ) (cid:18) Dη (cid:19) k · · · (cid:18) D n ηn ! (cid:19) k n (2.2) + ( D n w )( η )( Dη ) n . The condition µ ( a, w ) = p ensures that D k w ( a ) = D k w ( η (0)) = 0 , where k = 1 , , · · · , p − . Note that in (2.2), if k = n , then k + · · · + k n < n , and thus ( D k + ··· + k n w )( η (0)) = 0.Then(2.3) D n W (0) = 0 , where n = 1 , , · · · , p − . For n = p , we have(2.4) D p W (0) = ( D p w )( η (0))( η ′ (0)) p . Since η ′ (0) = | a | − D p w )( η (0)) = D p w ( a ) = 0 , we see that(2.5) D p W (0) = 0 . Hence, µ (0 , W ) = p easily follows from (2.3) and (2.5). (cid:3) Main results
Proof of Lemma 1.1.
Assume that w = u + iv is a sense-preserving harmonicself-mapping of D with µ (0 , w ) = p ≥
1. For any θ ∈ [0 , π ], let f be an analyticfunction of D , where Re f = u cos θ + v sin θ is harmonic in D . Then f ( D ) ⊂ S = { z ∈ C : | Re z | < } and f (0) = 0. If we write f = ξ + iϑ and w = h + ¯ g , then for z = x + iy ∈ D , ξ ( z ) = Re( w ( z ) e − iθ ) , and f ′ ( z ) = ξ x ( z ) − iξ y ( z )= h ′ ( z ) e − iθ + g ′ ( z ) e iθ . Therefore(3.1) D p f = D p he − iθ + D p ge iθ , oundary Schwarz lemma for harmonic mappings having zero of order p which shows that µ (0 , f ) = p , since µ (0 , w ) = p . Let δ = tan (cid:16) π f (cid:17) . Then by Lemma 2.2, we see that δ is an analytic function of D into D with µ (0 , δ ) = p .Applying Lemma A, we have | δ ( z ) | ≤ | z | p | z | + p ! | D p δ (0) | p ! | D p δ (0) || z | = | z | p | z | + π p ! | D p f (0) | π p ! | D p f (0) || z | , where the last equality holds since it follows from (2.1) that D p δ (0) = π D p f (0).On the other hand, let d ( z ) = e i π f ( z ) − e i π f ( z ) + 1 . Then d ( z ) = iδ ( z ). Using the following elementary inequalitytan 12 | Re ς | ≤ (cid:12)(cid:12)(cid:12)(cid:12) e iς − e iς + 1 (cid:12)(cid:12)(cid:12)(cid:12) , for all | Re ς | ≤ π , we see that tan (cid:18) (cid:12)(cid:12)(cid:12) Re π f (cid:12)(cid:12)(cid:12)(cid:19) ≤ | d | = | δ | . Thus(3.2) | Re f ( z ) | ≤ π arctan | δ | ≤ π arctan " | z | p | z | + π p ! | D p f (0) | π p ! | D p f (0) || z | . Using (3.1), we have (cid:12)(cid:12)(cid:12)(cid:12) D p f (0) p ! (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) D p h (0) e − iθ p ! + D p g (0) e iθ p ! (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) D p h (0) p ! (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) D p g (0) p ! (cid:12)(cid:12)(cid:12)(cid:12) = | a p | + | b p | . Elementary calculations show that for 0 ≤ r <
1, the function ϕ ( x ) = r + π x π xr is anincreasing function of x . These together with (3.2) show that(3.3) | u ( z ) cos θ + v ( z ) sin θ | ≤ π arctan (cid:20) | z | p | z | + π ( | a p | + | b p | )1 + π ( | a p | + | b p | ) | z | (cid:21) . The desired inequality (1.1) is now easy to follow, since | w ( z ) | = max θ ∈ [0 , π ] | ξ | = max θ ∈ [0 , π ] | u ( z ) cos θ + v ( z ) sin θ | . This completes the proof of Lemma 1.1. (cid:3)
Bai, Huang and Zhu
Proof of Theorem 1.1.
For any z ∈ D , since µ (0 , w ) = p , we see from Lemma1.1 that(3.4) | w ( z ) | ≤ π arctan (cid:20) | z | p | z | + π ( | a p | + | b p | )1 + π ( | a p | + | b p | ) | z | (cid:21) := M ( | z | ) . Since w is differential at z = 1, we know that w ( z ) = 1 + w z (1)( z −
1) + w ¯ z (1)(¯ z −
1) + ◦ ( | z − | ) . This together with (3.4) show that | w z (1)( z −
1) + w ¯ z (1)(¯ z −
1) + ◦ ( | z − | ) | ≤ M ( | z | ) . Therefore, 2Re[ w z (1)(1 − z ) + w ¯ z (1)(1 − ¯ z )] ≥ − M ( | z | ) + ◦ ( | z − | ) . Take z = r ∈ (0 ,
1) and letting r → − , it follows from M (1) = 1 that2Re[ w z (1) + w ¯ z (1)] ≥ lim r → − − M ( r )1 − r = 4 π ( p + 1) + π ( p − | a p | + | b p | )1 + π ( | a p | + | b p | ) . Then(3.5) Re[ w z (1) + w ¯ z (1)] ≥ π ( p + 1) + π ( p − | a p | + | b p | )1 + π ( | a p | + | b p | ) , hence the proof of the theorem is complete. (cid:3) Proof of Theorem 1.2.
For α ∈ T , let γ = η ( α ), where η ( z ) = ϕ a ( z ) = a − z − ¯ az .It is easy to see that γ ∈ T and η ( γ ) = α. Elementary calculations show that η ′ (0) = | a | − . For β ∈ T , let W ( ζ ) = ¯ βw ◦ η ( ζ γ ) = H + ¯ G , where ζ ∈ D . Then W ζ ( ζ ) = ¯ βw z ( η ( ζ γ )) η ′ ( ζ γ ) γ and W ¯ ζ ( ζ ) = ¯ βw ¯ z ( η ( ζ γ )) η ′ ( ζ γ )¯ γ. Using the following equation η ′ ( γ ) = − (1 − ¯ aα ) − | a | we have(3.6) Re( W ζ (1) + W ¯ ζ (1)) = Re ¯ β " w z ( α ) α | − ¯ aα | − | a | + w ¯ z ( α ) ¯ α | − ¯ aα | − | a | . oundary Schwarz lemma for harmonic mappings having zero of order p Since w is a sense-preserving harmonic self-mapping of D with µ ( a, w ) = p , it followsfrom Lemma 2.3 that W ( ζ ) is also sense-preserving in D with W ( D ) ⊂ D and µ (0 , W ) = p . Furthermore, we have W (0) = ¯ βw ( η (0)) = ¯ βw ( a ) = 0and W (1) = ¯ βw ( η ( γ )) = ¯ βw ( α ) = | β | = 1 . Using Theorem 1.1, we obtain the following inequality(3.7) Re( W ζ (1) + W ¯ ζ (1)) ≥ π ( p + 1) + π ( p − (cid:16) p ! | D p H (0) | + p ! | D p G (0) | (cid:17) π (cid:16) p ! | D P H (0) | + p ! | D P G (0) | (cid:17) . According to (2.4) and note that ¯ D p W (0) = ( ¯ D p w )( η (0)) (¯ η ′ (0)) p , we have(3.8) 1 p ! | D p H (0) | + 1 p ! | D p G (0) | = Λ ( p ) w ( a )(1 − | a | ) p , where Λ ( p ) w ( a ) = (cid:12)(cid:12)(cid:12) D p w ( a ) p ! (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ¯ D p w ( a ) p ! (cid:12)(cid:12)(cid:12) . It follows from (3.6), (3.7) and (3.8) thatRe (cid:18) ¯ β [ w z ( α ) α + w ¯ z ( α ) ¯ α ] (cid:19) ≥ π ( p + 1) + π ( p − ( p ) w ( a )(1 − | a | ) p π Λ ( p ) w ( a )(1 − | a | ) p − | a | | − aα | . If a = 0, thenRe (cid:18) ¯ β [ w z ( α ) α + w ¯ z ( α ) ¯ α ] (cid:19) ≥ π ( p + 1) + π ( p − | a p | + | b p | )1 + π ( | a p | + | b p | ) . This completes the proof of the theorem. (cid:3)
Acknowledgments . We would like to thank the anonymous referees for theirhelpful comments to improve this paper.
Funding . The research of the authors were supported by NNSF of China GrantNos. 11471128, 11501220, 11971124, 11971182, NNSF of Fujian Province GrantNos. 2016J01020, 2019J0101, Subsidized Project for Postgraduates’Innovative Fundin Scientific Research of Huaqiao University and the Promotion Program for Youngand Middle-aged Teacher in Science and Technology Research of Huaqiao University(ZQN-PY402).
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E-mail address : xiaojin [email protected] Jie Huang, School of Mathematical Sciences, Huaqiao University, Quanzhou362021, China and Technion-Israel institute of Technology, Guangdong Technion,241 Daxue Road, Shantou, Guangdong 515063, China.
E-mail address : [email protected]; Jian-Feng Zhu, School of Mathematical Sciences, Huaqiao University, Quanzhou362021, China.
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