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

1_{Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China} 2_{Department 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,

Δf4fzz 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

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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 Fαf 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

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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

Pαz

ba

iπ log

1zeiα

1−zeiα

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 inHp-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:

Vφ,Hz0

Fα

fz0 : f∈ S0

φ,

Vφ,Hz0, λ

Fα

fz0 : f∈ A0

φ, f0 λφ0 ,

VH,Sa,b,uz0

Fα

fz0 : f∈ Sa,b,u

,

VH,Sa,bz0, λ

Fα

fz0 : 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

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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 forVφ,Hz0 , Vφ,Hz0, λ ,VH,Sa,b,uz0 and VH,Sa,bz0, λ ,

respectively.

Theorem 1. The boundary∂Vφ,Hz0 ofVφ,Hz0 is the Jordan curve given by

−π, πθ−→φeiθz0

αφeiθ_z₀_. ₁₂

Proof. We defineVφz0 {fz0 :f ∈ S0φ }.In order to determine the setVφz0 ,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 in}_D_with_ω₀ _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

VBz0 {ωz0 :ω∈B0∩ S0 ∪ {0}} 13

coincides with the set{z:|z| ≤ |z0|}.Hence the region of variabilityVφz0 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 Vφ,Hz0 follows fromVφz0 . 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∂Vφ0,Hz0 ofVφ0,Hz0 is the Jordan curve given by

−π, πθ−→

1eiθ_z

0 1−eiθ_z

0

β

α

1eiθ_z

0 1−eiθ_z

0

β

−1−α. 15

Theorem 2. The boundary∂Vφ,Hz0, λ ofVφ,Hz0, λ is the Jordan curve given by

−π, πθ−→φz0δ

eiθz0, λ

α φz0δeiθz0, λ , 16

where

δcz, λ czλ

1czλ

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Proof. Letf ∈ A0 such thatf ≺ φfor someφ ∈ S0. Becausef ≺ φ, there exists a Schwarz functionωφ−1_◦_f _{∈ B0}_with_ω₀_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 Vφz0, λ . Then the determination of the set Vφ,Hz0, λ follows from

Vφz0, λ .Now, we define

Fωz ωz /z−

λ

1−λωz /z

, i.e., ωz zFωz λ

1Fωz λ

. ₁₉

We observe thatFω∈ B0.By the Schwarz lemma, we have|Fωz | ≤ |z|. If we set

Bλ

0

Fω:ω∈ B0, ω0 λ

20

then the region of variability{ωz0 :ω ∈ Bλ₀}coincides with the set{z:|z| ≤ |z0|}. It follows from the two expressions in19 thatVφz0, 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∂Vφ,Hz0,0 ofVφ,Hz0,0 is the Jordan curve given by

−π, πθ−→φz2₀eiθα φz2

0eiθz0 . 22

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

Vφ0,Hz0, λ

Fα

fz0 :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/β1. 24

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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∂Vφ0,Hz0, λ ofVφ0,Hz0, λ is the Jordan curve given by

−π, πθ−→

1z0δeiθz0, λ 1−z0δeiθz0, λ

β

α

1z0δeiθz0, λ 1−z0δeiθz0, λ

β

−1−α, 25

whereφ0z andδcz, λ are given by14 and17 , respectively.

The boundary∂Vφ0,Hz0,0 ofVφ0,Hz0,0 is the Jordan curve given by

−π, πθ−→

1z2 0eiθ 1−z2

0eiθ

β

α

1z2 0eiθ 1−z2

0eiθ

β

−1−α. 26

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

−π, πθ−→ ab iπ

⎡ ⎣log

1−z0eiθe−2πai/ab 1−z0eiθ

−αlog

1−z0eiθe−2πai/ab 1−z0eiθ

⎤ ⎦. 27

Proof. We define VSa,b,uz0 {fz0 : f ∈ Sa,b,u}. It suﬃces to determineVSa,b,uz0 as the

region of variabilityVH,Sa,b,uz0 follows fromVSa,b,uz0 . In order to do this, first we consider

Tz ab

iπ logwz , wz

1−ze−2πai/ab

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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

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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

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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

Vω,uz0 {ωz0 :ω∈B0∩ S0 ∪ {0}} 30

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

the set

ab

iπ log

1−ze−2πai/ab

1−z

:|z| ≤ |z0|

. 31

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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∂VH,Sa,bz0, λ ofVH,Sa,bz0, λ is the Jordan curve given by

−π, πθ−→ ab iπ

log

1−z0δ

z0eiθ, λ

e−2πai/ab

1−z0δ

z0eiθ, λ

−αlog

1−z0δ

z0eiθ, λ

e−2πai/ab

1−z0δ

z0eiθ, λ

⎤ ⎦,

32

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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/ab _{and consider}

VSa,bz0, λ

fz0 :f∈ Sa,b, f0 p

1−qλ. 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◦gz e

fz/p₋₁

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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

ωfz /z−λ

1−λωfz /z

. 36

Clearly,Qf ∈ B0. If we let

Sa,b,ωf,λ

Qf : ωf ∈ B0, ωf0 λ

,

VQfz0

ωfz0 : ωf ∈Sa,b,ωf,λ

,

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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 variabilityVQfz0 coincides

with the set{z:|z| ≤ |z0|}. Equation36 implies that

ωfz

zQfz λ

1Qfz 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 , and

27

Table 1gives the list of these parameter values corresponding to Figures1–8which concern the regions of variability for∂Vφ0,Hz0 ,∂Vφ0,Hz0,0 , and∂VH,Sa,a,uz0 , respectively.

Using Mathematicasee13 , we describe the boundary sets∂Vφ0,Hz0 ,∂Vφ0,Hz0,0 , and∂VH,Sa,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, ∂Vφ0,Hz0 ,∂Vφ0,Hz0,0 , and∂VH,Sa,a,uz0 , whereas theβvalue is applicable only for the first

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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

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