• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

    2015-10-13 01:59:52MengkunZhuandXinzhongHuang
    Journal of Mathematical Study 2015年3期

    Mengkun Zhuand Xinzhong Huang

    SchoolofMathematicalScience,HuaqiaoUniversity,Quanzhou362021,Fujian Province,P.R.China.

    The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

    Mengkun Zhu?and Xinzhong Huang

    SchoolofMathematicalScience,HuaqiaoUniversity,Quanzhou362021,Fujian Province,P.R.China.

    .Some sharp estimates for coefficients,distortion and the growth order are obtained for harmonic mappingsf∈TLαHwhich are locally univalent harmonic mappings in the unit diskD={z:|z|< 1}.Moreover,denoting the subclassTSαHof the normalized univalent harmonic mappings,we also estimate the growth of|f|,f∈TSαH,and their covering theorems.

    AMS subject classifications:30D15,30D99

    Harmonic mapping,coefficient estimate,distortion theorem,covering problem.

    1 Introduction

    A complex-valued harmonic functionfin the unit diskDhas a canonical decomposition

    wherehandgare analytic inDwithg(0)=0.Usually,we callhthe analytic part offandgthe co-analytic part off.A complete and elegant account of the theory of planar harmonic mappings is given in Duren,s monograph[1].

    In[2],Ikkei Hotta and Andrzej Michalski denoted the classLHof all normalized locally univalent and sense-preserving harmonic functions in the unit disk withh(0)=g(0)=h′(0)?1=0.Which means every functionf∈LHis uniquely determined by coefficients of the following power series expansions

    wherean,bn∈C,n=2,3,4,...Clunie and Sheil-small introduced in[3]the classSHof all normalized univalent harmonic mappings inD,obviously,SH?LH.

    Lewy[4]proved that a necessary and sufficient condition forfto be locally univalent and sense-preserving inDisJf(z)>0,where

    To such a functionf,not identically constant,let

    thenω(z)is analytic inDwith|ω(z)|< 1,it is called the second complex dilatation off.

    In[5],Silverman investigated the subclass ofTwhich denoted byT?(β),starlike of orderβ(0≤β<1).That is,a functionF(z)∈T?(β)if Re{zF′(z)/F(z)}>β,z∈D.It was proved in[5]that

    Corollary 1.1.

    In[7-8],Dominika Klimek and Andrzej Michalski studied the cases when the analytic partshis the identity mapping or a convex mapping,respectively.The paper[2]was devoted to the case when the analytichis a starlike analytic mapping.In[9],Qin Deng got sharp results concerning coefficient estimate,distortion theorems and covering theorems for functions inT.The main idea of this paper is to characterize the subclasses ofLHandSHwhenh∈T.

    In order to establish our main results,we need the following theorems and lemmas.

    Lemma 1.1.([10])If f(z)=a0+a1z+...+anzn+...is analytic and|f(z)|≤1onD,then

    Theorem 1.2.([8])If f∈T,then

    with equality for

    Theorem 1.3.([8])If f∈T,then

    with equality for

    2 Main results and their proofs

    Similar with the papers[2,7,8]and[12],we consider the following function sets.Definition 2.1.Forα∈[0,1),let

    Definition 2.2.Forα∈[0,1),let

    Forf∈TLαH,applying Theorem 1.1 and Lemma 1.1,we can prove the following theorem.

    Theorem 2.1.If f∈TLαH,then|an|≤1/n,n=2,3,...,and

    It is sharp estimate for|b2|,the extremal functions are

    Moreover,

    then|an|≤ 1/nby Theorem 1.1.Letg′(z)=ω(z)h′(z),whereω(z)is the dilatation off.

    Sinceω(z)is analytic in D,it has a power series expansion

    wherecn∈C,n=0,1,2,...,and|c0|=|ω(0)|=|g′(0)|=|b1|=α.Recall that|ω(z)|<1 for allz∈D,then by Lemma 1.1,we have

    Together with the formula(1.3),(1.5)and(2.4),we give

    Hence,we obtain

    Applying the facts(1.6)and(2.5),and by simple calculation,we have

    Applying the formula(1.5),we obtain

    which implies the estimate of(2.1)is sharp.Sinceg0(0)=0,by integration,we uniquely deduce

    Obviously,|ω0(z)|<1,z∈D,which meansf0(z)=h0(z)+g0(z)∈TLαH.

    In the same way,

    Hence,the proof is completed.

    Corollary 2.1.If

    Proof.By simple calculation,we have

    then the corollary follows immediately from Theorem 2.1.

    Since the analytic parthoff∈TLαHbelongs toT,we have the following distortion estimate ofhby Theorem 1.2[9]

    Our next aim is to give the distortion estimate of the co-analytic partgoff∈TLαH.

    These inequalities are sharp.The equalities hold for the harmonic function f0(z)which is defined in(2.2).

    Proof.Letb1=g′(0)=αeiψ.Consider the function

    It satisfies assumptions of the Schwarz lemma,which gives

    It is equivalent to

    and the equality holds only for the functions satisfying

    where?∈R.Hence,applying the triangle inequalities and the formula(2.11)we have

    Finally,applying the formula(2.6)together with(2.13)to the identityg′=ωh′,we obtain(2.7)and(2.8).The functionf0(z)defined in(2.2)shows that inequalities(2.7)and(2.8)are sharp.The proof is completed.

    Proof.Letαtend to 1 in the estimate(2.8),then the corollary follows from theorem 2.2 immediately.

    From the Theorem 1.3[9],we can get the growth estimate of the analytic parthoff∈TLαH

    Next results,we give the growth estimate of co-analytic partgoff∈TLαH.

    The inequality is sharp.The equality hold for the harmonic function f0(z)which is defined in(2.2).

    Proof.Let Γ:=[0,z],applying the estimate(2.8)we have

    By integration,we obtain the estimate(2.15).The functionf0(z)defined(2.2)shows that the inequality(2.15)is sharp.

    Using the distortion estimates in Theorem 1.2[9]and Theorem 2.2,we can easily deduce the following Jacobian estimates off∈TLαH.

    Proof.Observe that iff∈TLαH,thenh′does not vanish inD.We can give the Jacobian offin the form

    whereωis the dilatation off.Applying(2.6)and(2.13)to the(2.17)we obtain

    and

    this completes the proof.

    Since every univalent function is locally univalent,we can give the growth estimate off∈TSαH.

    Proof.For any pointz∈Dand supposer:=|z|,we denoteDr:=D(0,r)={z∈D:|z|<r},and let

    Byg′=ωh′and the formula(2.6)and(2.13),we obtain

    Hence,we have

    Integrating,we obtain the estimate(2.19).To prove(2.20)we simply use the inequality

    Then,by the formula(1.8)and(2.15)with simple calculation we have(2.20),this completes the proof.

    Finally,the growth estimate off∈TSαHyields a covering estimate.

    The images of α∈[0,1)■→R are shown in Figure 1.

    Proof.If we let|z|tend to 1 in the estimate(2.19),then the Theorem 2.6 follows immediately from the argument principle for harmonic mappings.

    Figure 1:The image of

    Acknowledgments

    The authors are grateful to the referees for their useful comments and suggestion.This work is partially supported by NNSF of China(11101165),the Natural Science Foundation of Fujian Province of China(2014J01013),NCETFJ Fund(2012FJ-NCET-ZR05),Promotion Program for Young and Middle-aged Teacherin Science and Technology Research of Huaqiao University(ZQN-YX110).

    [1]P.Duren.Harmonic mappings in the plane.Cambridge Univ.Press,Cambridge,2004.

    [2]I.Hotta and A.Michalski.Locally one-to-one harmonic functions with starlike analytic part.Preprint,available from http://arxiv.org/abs/1404.1826.

    [3]J.G.Clunie and T.Sheil-Small.Harmonic univalent functions.Ann.Acad.Sci.Fenn.Ser.A.I.Math.9:3-25,1984.

    [4]H.Lewy.On the non-vanishing of the Jacobian in certain one-to-one mappings.Bull.Amer.Math.Soc.,42:689-692,1936.

    [5]H.Silverman.Univalent functions with negative coefficients.Proc.Amer.Math.Soc.,51(1):109-116,1975.

    [6]G.Karpuzov,Y.Sibel,M.¨Ozt¨urk,and M.Yamankaradeniz.Subclass of harmonic univalent functions with negative coefficients.Appl.Math.Comput.,142(2-3):469-476,2003.

    [7]D.Klimek,A.Michalski.Univalent anti-analytic perturbation of the identity in the unit disc.Sci.Bull.Chelm.,1:67-78,2006.

    [8]D.Klimek and A.Michalski.Univalent anti-analytic perturbation of convex conformal mapping in the unit disc.Ann.Univ.Mariae Curie-Sklodowska Sect.A,61:39-49,2007.

    [9]Qin Deng.On univalent functions with negative coefficients.Appl.Math.and Comput.,189:1675-1682,2007.

    [10]I.Graham and G.Kohr.Geometric Function Theory in One and Higher Dimensions.Marcel Dekker Inc,New York,2003.

    [11]X.Z.Huang.Estimates on Bloch constants for planar harmonic mappings.J.Math.Anal.Appl,337(2):880-887,2008.

    [12]S.Kanas and D.Klimek.Harmonic mappings pelated to functions with bounded boundary rotation and norm of the pre-schwarzian derivative.Bull.Korean Math.Soc.51(3):803-812,2014.

    [13]P.Duren.Univalent functions.Springer-Verlag,Berlin-New York,1975.

    22 May,2015;Accepted 9 July,2015

    ?Corresponding author.Email addresses:ZMK900116@163.com(M.Zhu),huangXZ@hqu.edu.cn(X.Huang)

    玉林市| 尼木县| 宣汉县| 安远县| 罗田县| 岚皋县| 南汇区| 岑溪市| 临城县| 股票| 义乌市| 石棉县| 尚义县| 池州市| 广汉市| 漾濞| 达州市| 孝感市| 三门县| 阳信县| 泰安市| 慈溪市| 邹城市| 杭锦后旗| 余江县| 普兰县| 松江区| 海安县| 铁岭市| 海盐县| 新乡县| 阳城县| 敦化市| 赣州市| 西平县| 藁城市| 改则县| 巩留县| 田林县| 杭锦旗| 修文县|