General Approach to Regions of Variability via Subordination of Harmonic Mappings

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Volume 2009, Article ID 736746,15pages doi:10.1155/2009/736746

Research Article

General Approach to Regions of Variability via Subordination of Harmonic Mappings

Sh. Chen,

¹

S. Ponnusamy,

²

and X. Wang

¹

1Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China

2Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India

Correspondence should be addressed to X. Wang,[email protected] Received 17 October 2009; Accepted 20 November 2009

Recommended by Narendra Kumar Govil

Using subordination, we determine the regions of variability of several subclasses of harmonic mappings. We also graphically illustrate the regions of variability for several sets of parameters for certain special cases.

Copyrightq2009 Sh. Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Introduction

A planar harmonic mapping in a simply connected domain D ⊂ C is a complex-valued function f uiv defined in D for which bothu andv are real harmonic inD, that is, Δf4f_zz 0,whereΔrepresents the Laplacian operator. The mappingfcan be written as a sum of an analytic and antianalytic functions, that is,f hg.We refer to1and the book of Duren2for many interesting results on planar harmonic mappings.

We note that the compositionf◦φof a harmonic functionfwith an analytic functionφ is harmonic, but this is not true for the functionφ◦f, that is, an analytic function of a harmonic function need not be harmonic. It is known that2, Theorem 2.4the only univalent harmonic mappings ofContoCare the aﬃne mappingsgz βzγzη|β|/|γ| .Motivated by the work of3, we say thatFis an aﬃne harmonic mapping of a harmonic mapping offif and only ifFhas the form

F :Fα

f

fαf 1

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for some α ∈ C with |α| < 1. Obviously, an aﬃne transformation applied to a harmonic mapping is again harmonic. The aﬃne harmonic mappings F_αf and f share many properties in commonsee4 .

LetHdenote the class of analytic functions in the unit diskD{z∈C: |z|<1}, and A0 {h∈ H : h0 0}. Also, letS0 be the subclass ofA0 consisting of functions that are univalent inD. For a givenφ ∈ S0, we will denote byA0φ andS0φ the subsets defined by{h ∈ A0 : h≺ φ}and{h ∈ S0 :h ≺ φ} ∪ {0}, respectively. From now onwards, we use the notation f ≺ g, or, fz ≺ gz inD for analytic functionsf and g on Dto mean the subordination, namely there existsω ∈ B0 such thatfz gωz . Here B0 denotes the class of analytic mapsψ of the unit diskDinto itself with the normalizationψ0 0. We remark that ifgis univalent inD, then the subordinationf≺gis equivalent to the condition thatf0 g0 andfD ⊂gD . This fact will be used in our investigation. Moreover, the special choices ofφhave been the subjects of extensive studies; we suggest that the reader to consult the books of Pommerenke5, Duren6and of Miller and Mocanu7for general back ground material.

We denote byAa,b the class of functions f ∈ H with f0 b−a /2, and −a <

Refz < bforz∈D. We note that ifa >0, then each functionf ∈ Aa,aobviously satisfy the normalization conditionf0 0. A functionf∈ His called a Bloch function if

f

Bsup

z∈D

1− |z|²fz <∞. 2

Then the set of all Bloch functions forms a complex Banach spaceBwith the norm · given by

ff0 f_B, 3 see8. Every bounded function inHis Bloch, but there are unbounded Bloch functions, as can be seen also from the following result which shows thatAa,b⊂ B.

Proposition 1. Iff∈ Aa,b, then f _B≤2ba /π.The constant 2ba /πis sharp. In particular, iff∈ Aa,athen f _B≤4a/πand the constant 4a/πis sharp.

Proof. Let

Pz ba iπ log

1z

1−z b−a

2 , z∈D. 4

ThenP0 b−a /2,

Pz 2ba

iπ1−z² 5

andPmapsDunivalently onto the vertical strip{w:−a <Re w < b},and P _B2ba /π.

Consequently, iff∈ Aa,b, then we havef≺Pand so, there exists a Schwarz functionω∈ B0

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such thatfz Pωz . Thus, asω0 0, the Schwarz-Pick lemma gives that

1− |z|²fz

1− |z|²ωz Pωz ≤

1− |ω|²Pω ≤ P _B 6 so that f _B≤2ba /π,with equality forfz Pαz , where

P_αz ba iπ log

1ze^iα 1−ze^iα

b−a

2 , α∈R. 7

It may be interesting to remark that the functionfz _∞

n1z²ⁿ belongs to B 9, Theorem 1is a good example of a Bloch function which is not inH^p-space for anyp.Bloch functions are intimately close with univalent functionssee5 .

In order to state our main results, we introduce some basics. For givena, b >0, letSa,b

be the class of functionsf∈ A0and−a <Re fz < bforz∈D. Now, we define Sa,b,u

f :f∈ Sa,b andf is univalent

∪ {0}. 8

We note that each function inSa,bhas the normalizationf0 0.For any fixedz₀ ∈D\ {0}

andλ∈Cwith 0<|λ|<1,we consider the following sets:

V_φ,Hz0 Fα

f

z0 : f∈ S0

φ , V_φ,Hz0, λ

F_α f

z0 : f∈ A0

φ

, f0 λφ0 , V_H,S_a,b,uz0

Fα

f

z0 : f∈ Sa,b,u

,

V_H,S_a,bz0, λ

Fα

f

z0 : f0 λba iπ

1−e^−2πai/ba

, f∈ Sa,b

.

9

We now recall the definition of subordination for the harmonic case from10, page 162. Let f andF be two harmonic functions defined onD. We say f is subordinate toF, denoted byf ≺F, iffz Fωz , whereω∈ B0. Obviously, iff₁andf₂are two harmonic functions inD, then

f1 ≺f2⇐⇒Fα

f1

≺Fα

f2

. 10

Here we see thatαis the analytic dilatation for bothFαf1 andFαf2 .

For each fixedz₀ ∈D, using extreme function theory, it has been shown by Grunsky see, e. g., Duren6, Theorem 10.6 that the region of variability of

V_Sz0

logfz0

z0

:f∈ S

11 is precisely a closed disk, where S {f ∈ S0:f0 1}. Recently, by using the Herglotz representation formula for analytic functions, many authors have discussed region

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of variability problems for a number of classical subclasses of univalent and analytic functions in the unit disk Dsee11,12 and the references therein . Because the class of harmonic univalent mappings includes the class of conformal mappings, it is natural to study the class of harmonic mappings. In the following, we will use the method of subordination and determine the regions of variability forV_φ,Hz0 , V_φ,Hz0, λ ,V_H,S_a,b,uz0 and V_H,S_a,bz0, λ , respectively.

Theorem 1. The boundary∂V_φ,Hz0 ofV_φ,Hz0 is the Jordan curve given by

−π, πθ−→φ e^iθz0

αφ e^iθz0

. 12

Proof. We defineV_φz0 {fz0 :f ∈ S0φ }.In order to determine the setV_φz0 ,we first recall that eachf ∈ S0φ \ {0}can be written asfz φωz for someω ∈ B0\ {0}. By the Riemann mapping theorem,ω φ⁻¹◦f is univalent and analytic inDwithω0 0.

It follows from the classical Schwarz lemma that for anyω ∈ B0,we have|ωz | ≤ |z|inD.

Because, in our situationωis also univalent inD, we easily show that the region of variability V^Bz0 {ωz0 :ω∈B0∩ S0 ∪ {0}} 13 coincides with the set{z:|z| ≤ |z0|}.Hence the region of variabilityV_φz0 is precisely the set {φz :|z| ≤ |z0|}.We remark thatVφz0 depends only on|z0|, becauseS0 is preserved under rotation and therefore, we may assume that 0< z0 <1. Finally, the region of variability V_φ,Hz0 follows fromV_φz0 . The proof of this theorem is complete.

There are many choices forφ for whichTheorem 1 is applicable. For example, if we chooseφto be

φz 1z

1−z

β−1, 14

for some 0< β≤2, then we have following result fromTheorem 1.

Corollary 1. The boundary∂V_φ₀_,Hz0 ofV_φ₀_,Hz0 is the Jordan curve given by

−π, πθ−→

1e^iθz₀ 1−e^iθz₀

β

α

1e^iθz₀ 1−e^iθz₀

β

−1−α. 15

Theorem 2. The boundary∂V_φ,Hz0, λ ofV_φ,Hz0, λ is the Jordan curve given by

−π, πθ−→φ z₀δ

e^iθz₀, λ

α φz0δe^iθz₀, λ , 16

where

δcz, λ czλ 1czλ

c∈D

. 17

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Proof. Letf ∈ A0 such thatf ≺ φfor someφ ∈ S0. Becausef ≺ φ, there exists a Schwarz functionωφ⁻¹◦f ∈ B0withω0 f0 /φ0 λ,where|λ| ≤1.Therefore, for any fixed z₀∈D\ {0}andλ∈Cwith 0<|λ| ≤1, it is natural to consider the set

Vφz0, λ

fz0 :f∈ A0

φ

, f0 λφ0

. 18

First, we determine V_φz0, λ . Then the determination of the set V_φ,Hz0, λ follows from Vφz0, λ .Now, we define

F_ωz ωz /z−λ 1−

λωz /z, i.e., ωz zFωz λ

1Fωz λ . 19

We observe thatF_ω∈ B0.By the Schwarz lemma, we have|Fωz | ≤ |z|. If we set B^λ₀

F_ω:ω∈ B0, ω0 λ

20

then the region of variability{ωz0 :ω ∈ B^λ₀}coincides with the set{z:|z| ≤ |z0|}. It follows from the two expressions in19 thatV_φz0, λ coincides with the set

φz0δz, λ :|z| ≤ |z0|, whereδz, λ zλ 1zλ

. 21

The proof of this theorem is complete.

The caseλ0 ofTheorem 2gives the following result.

Corollary 2. The boundary∂V_φ,Hz0,0 ofV_φ,Hz0,0 is the Jordan curve given by

−π, πθ−→φ z²₀e^iθ

α φz²₀e^iθz₀ . 22

Ifφ0z is given by14 for some 0< β≤2, thenφ₀0 2βandVφ0,Hz0, λ reduces to V_φ₀_,Hz0, λ

F_α f

z0 :f∈ A0

φ₀

, f0 2βλ

23

and the correspondingωz in the proof of the theorem will be precisely of the form

ωz

1fz 1/β −1 1fz _1/β

1. 24

This observation gives the following corollary.

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−0.2

−0.1 0.1 0.2

−0.2 0.2 0.4

a

−0.02

−0.01 0.01 0.02

−0.04 −0.02 0.02 0.04

b

−6

−4

−2 2 4 6

−10 −5 5 10

c Figure 1

Corollary 3. The boundary∂V_φ₀_,Hz0, λ ofV_φ₀_,Hz0, λ is the Jordan curve given by

−π, πθ−→

1z₀δe^iθz₀, λ 1−z₀δe^iθz₀, λ

_β α

1z₀δe^iθz₀, λ 1−z₀δe^iθz₀, λ

_β

−1−α, 25

whereφ₀z andδcz, λ are given by14 and17 , respectively.

The boundary∂V_φ₀_,Hz0,0 ofV_φ₀_,Hz0,0 is the Jordan curve given by

−π, πθ−→

1z²₀e^iθ 1−z²₀e^iθ

β

α

1z²₀e^iθ 1−z²₀e^iθ

β

−1−α. 26

Theorem 3. The boundary∂V_H,S_a,b,uz0 ofV_H,S_a,b,uz0 is the Jordan curve given by

−π, πθ−→ ab iπ

⎡

⎣log

1−z0e^iθe^−2πai/ab 1−z0e^iθ

−αlog

1−z0e^iθe^−2πai/ab 1−z0e^iθ

⎤

⎦. 27

Proof. We define V_S_a,b,uz0 {fz0 : f ∈ Sa,b,u}. It suﬃces to determineV_S_a,b,uz0 as the region of variabilityV_H,S_a,b,uz0 follows fromV_S_a,b,uz0 . In order to do this, first we consider

Tz ab

iπ logwz , wz 1−ze^−2πai/ab

1−z . 28

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−1 1 2 3

2 4 6 8

a

−0.5 0.5 1

−1 1 2 3

b

−60

−40

−20 20 40 60

−75 −50 −25 25 50 75

c Figure 2

−0.6

−0.4

−0.2 0.2 0.4

−0.5 −0.25 0.25 0.5 0.75 1

a

−0.3

−0.2

−0.1 0.1 0.2

−0.2 0.2 0.4

b

−30

−20

−10 10 20 30

−30 −20 −10 10 20 30

c Figure 3

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−1.5

−1

−0.5 0.5 1 1.5

a

−0.4

−0.2 0.2 0.4

−0.2−0.1 0.1 0.2 0.3

b

−10

−5 5 10

−7.5 −5 −2.5 2.5 5 7.5

c Figure 4

Then T0 0. We see that the M ¨obius transformationwz maps the open unit disk D conformally onto the half-plane

wuiv:usin πa ab

vcos πa ab

>0

29

and so, we easily obtain that T maps D conformally onto the vertical strip {w : −a <

Re w < b}. This observation shows thatT ∈ Sa,b,uand is in fact an extremal function for this class.

Next, we choose an arbitraryf∈ Sa,b,u\ {0}. Then we havef≺Tand so, there exists a Schwarz functionω∈ B0\{0}such thatfz Tωz .Note that bothfandTare univalent inDand so,ωT⁻¹◦fis univalent inDwithω0 0. It follows from the classical Schwarz

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−0.1

−0.05 0.05 0.1

−0.1 −0.05 0.05 0.1

a

−0.004

−0.002 0.002 0.004

−0.004 −0.002 0.002 0.004

b

−2

−1 1 2

−3 −2 −1 1 2 3

c Figure 5

lemma that|ωz | ≤ |z|inD. Becauseω is also univalent inD, we obtain that the region of variability of

V_ω,uz0 {ωz0 :ω∈B0∩ S0 ∪ {0}} 30

coincides with the set{z:|z| ≤ |z0|}.Hence the region of variabilityV_S_a,b,uz0 coincides with the set

ab iπ log

1−ze^−2πai/ab 1−z

:|z| ≤ |z0|

. 31

The proof ofTheorem 3is complete.

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−0.4

−0.2 0.2 0.4 0.6

−0.2−0.1 0.1 0.2 0.3

a

−0.3

−0.2

−0.1 0.1 0.2 0.3 0.4

−0.1 0.1 0.2

b

−150

−100

−50 50 100 150

−75 −25 25 75

c Figure 6

Theorem 4. The boundary∂V_H,S_a,bz0, λ ofV_H,S_a,bz0, λ is the Jordan curve given by

−π, πθ−→ ab iπ

log

1−z0δ

z0e^iθ, λ

e^−2πai/ab 1−z0δ

z0e^iθ, λ

−αlog

1−z₀δ

z₀e^iθ, λ

e^−2πai/ab 1−z₀δ

z₀e^iθ, λ

⎤

⎦,

32

whereδcz, λ is given by17 .

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−0.4

−0.2 0.2 0.4

−0.5 −0.25 0.25 0.5 0.75 1

a

−0.3

−0.2

−0.1 0.1 0.2 0.3

−0.4 −0.2 0.2 0.4 0.6

b

−100

−50 50 100

−60−40 −20 20 40 60

c Figure 7

Proof. For convenience, we letp ab /iπ andqe^−2πai/aband consider

V_S_a,bz0, λ

fz0 :f∈ Sa,b, f0 p 1−q

λ

. 33

As before, it suﬃces to prove the theorem forV_S_a,bz0, λ . Letf ∈ Sa,bwithf0 p1−q λ.

Define

gz fz

p , hz e^z, φz z−1

z−q. 34

Then, by the mapping properties of these functions, it can be easily seen that the composed mapping

ωfz

φ◦h◦g

z e^f^{z /p}−1

e^{fz /p}−q 35

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−0.4

−0.2 0.2

−0.2 0.2 0.4

a

−0.2

−0.1 0.1 0.2

−0.2 −0.1 0.1 0.2

b

−30

−20

−10 10 20 30

−30 −20 −10 10 20 30

c Figure 8

is analytic inDand maps unit diskDintoDsuch that ωf0 0 andω_f0 λ. Next, we introduceQf :D → Dby

Qfz ω_fz /z−λ 1−λ

ω_fz /z. 36

Clearly,Q_f ∈ B0. If we let

S_a,b,ω_f_,λ

Q_f : ω_f ∈ B0, ω_f0 λ , V_Q_fz0

ω_fz0 : ω_f ∈S_a,b,ω_f_,λ ,

37

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Table 1: Subordination of harmonic mappings.

Figure z0 α β ab

1 0.09289160.0656754i 0.3433080.551846i 1.25961 58.9326

2 −0.495149−0.48309i 0.3774740.363979i 1.14901 81.0473 3 −0.210195−0.485306i 0.126883−0.247013i 0.57185 45.4015 4 −0.1172780.329628i −0.1830410.337725i 1.44013 21.5077

5 0.03707620.00949962i 0.0147993−0.00392657i 1.42167 59.4649

6 0.3123030.721208i −0.5242270.716229i 0.187546 82.7409

7 0.315822−0.788402i 0.0532365−0.057638i 0.276943 71.5991

8 −0.6608990.013848i −0.0571237−0.691304i 0.234536 31.2565

then, by the Schwarz lemma, we have|Qfz | ≤ |z|. The region of variabilityV_Q_fz0 coincides with the set{z:|z| ≤ |z0|}. Equation36 implies that

ω_fz z

Q_fz λ

1Q_fz λ . 38

It follows from35 and38 thatV_S_a,bz0, λ coincides with the set

plog1−z0δz, λ q

1−z0δz, λ :|z| ≤ |z0|, whereδz, λ zλ 1zλ

. 39

The proof ofTheorem 4is complete.

Geometric View of the Jordan Curves:15 ,26 , and27

Table 1gives the list of these parameter values corresponding to Figures1–8which concern the regions of variability for∂V_φ₀_,Hz0 ,∂V_φ₀_,Hz0,0 , and∂V_H,S_a,a,uz0 , respectively.

Using Mathematicasee13 , we describe the boundary sets∂V_φ₀_,Hz0 ,∂V_φ₀_,Hz0,0 ,

and∂V_H,S_a,a,uz0 described by the Jordan curve given by15 ,26 , and27 , respectively. In

the program below, “z0 stands forz₀,” “Alphaforα,” and “Betaforβ.’’

InTable 1, the parameter values ofz0andαare common for all the three cases, namely,

∂V_φ₀_,Hz0 ,∂V_φ₀_,Hz0,0 , and∂V_H,S_a,a,uz0 , whereas theβvalue is applicable only for the first two cases and theabvalues listed in the last column is meant only for the last case.

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(∗ Geometric view the main Theorem.... ∗) Remove["Global‘∗"];

z0 = Random [] Exp[I Random[Real, {-Pi, Pi}]]

\[Alpha] = Random[]Exp[I Random[Real, {-Pi, Pi}]]

\[Beta] = Random[Real, {0, 2}]

a = Random[Real, {0,100}]

Print["z0=", z0]

Print["\[Alpha]=", \[Alpha]]

Print["\[Beta]=", \[Beta]]

Print["a", a]

myf1[the , \[Alpha] , \[Beta] , z0 ]:=

((1+Exp[I∗the]∗z0)/(1-Exp[I∗the]∗z0))\[Beta] +

\[Alpha]∗Conjugate[((1+Exp[I∗the]∗z0)/(1-Exp[I∗the]∗z0))\[Beta]]

- 1-\[Alpha];

myf2[the , \[Alpha] , \[Beta] ,z0 ]:=

((1+Exp[I∗the]∗z0∗z0)/(1-Exp[I∗the]∗z0∗z0))\[Beta]+

\[Alpha]∗Conjugate[((1+Exp[I∗the]∗z0∗z0)/(1-Exp[I∗the]∗z0∗z0))\[Beta]]

-1-\[Alpha];

myf3[the , \[Alpha] , a ,z0 ]:=

(2a)/(I∗Pi)(Log((1-Exp[I∗the]∗Exp[-I∗Pi]∗z0)/(1-Exp[I∗the]∗z0))-

\[Alpha]∗Conjugate[Log((1-Exp[I∗the]∗Exp[-I∗Pi]∗z0)/(1-Exp[I∗the]∗z0))]) image1 = ParametricPlot[{Re[myf1[the, \[Alpha], \[Beta], z0]],

Im[myf1[the, \[Alpha], \[Beta], z0]]}, {the, -Pi, Pi},

AspectRatio -> Automatic,DisplayFunction -> $DisplayFunction, TextStyle ->{FontFamily -> "Times", FontSize -> 14},

AxesStyle ->{Thickness[0.0035]}];

image2 =ParametricPlot[{Re[myf2[the, \[Alpha], \[Beta], z0]], Im[myf2[the, \[Alpha], \[Beta], z0]]}, {the, -Pi, Pi},

AspectRatio -> Automatic,DisplayFunction ->$DisplayFunction, TextStyle ->{FontFamily -> "Times", FontSize -> 14},

image3 =ParametricPlot[{Re[myf3[the, \[Alpha], a, z0]], Im[myf3[the, \[Alpha], a, z0]]}, {the, -Pi, Pi},

AspectRatio -> Automatic,DisplayFunction -> $DisplayFunction, TextStyle ->{FontFamily -> "Times", FontSize -> 14},

Clear[the, z0, \[Alpha], \[Beta], a, myf1, myf2, myf3];

Acknowledgment

The research was partly supported by NSFs of chinaNo. 10771059 .

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