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Uniqueness of Meromorphic Functions Sharing One Set

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Uniqueness of Meromorphic Functions Sharing One Setof,Of,One,Set,one,set

    Uniqueness of Meromorphic Functions

    Sharing One Set

    COMMUNICATIONS

    IN

    MATHEMATICAL

    RESEARCH

    26(4)(2010),353-360

    UniquenessofMeromorphicFunctionsSharing

    OneSet

    QIJIAN-MINGANDYIHONG-XUN

    (Scho.fofMathem.tics,ShandongUniversity,Jinan,250100)

    CommunicatedbyJiYouqing

    Abstract:Inthispaper,westudyuniquenessofmeromorphicfunctionssharingone set,andobtainsomeresults,whichimproveandextendtheoriginalresults. Keywords:meromorphicfunction,Nevanlinnatheory,uniqueness,sharingvalue 2000MRsubjectclassification:3OD35.3OD45

    DocumentCOde:A

    ArticleID:1674.5647(2010)04-0353.08

    1Introduction

    Inthiswork,bymeromorphicfunctionwealwaysmeanameromorphicfunctioninthe complexplaneC.Weassumethatthereaderisfamiliarwiththestandardnotationofvalue distributiontheor~andthiscanbefound,forinstance,in[1or[2].

    Let,beanonconstantmeromorphicfunction.Weuse?(《?)todenotethecount

    ingfunct

    .

    ionforthezerosoffwithmultiplicity?kandcountingmultiplicities,whileby

    Nk)(r,?)thecountingfunctionofthezerosof,withmultiplicities?kandcounting

multiplicities.Wedenotebys(r,f)anyfunctionsatisfying

    S(r,f)=o(T(r,,)),r.-??.

    possiblyoutsideasetoffinitemeasure.Define

    (,

    

    LetSbeasetofcomplexnumbers.Define

    E(S,,)=

    Receiveddate:May18,2009.

    Foundationitem:TheNSF(10771121)ofChina,theNSF(Z2008A01)ofShandongandtheRFDP

    (20060422049).

    ,,0

    =

    

    

    

    Z

    ,?L

    u

    354C0MM.MATH.RESVOL.26

    whereeachzerooff(z)awithmultiplicitymiscountedmtimesinE(S,,).Thenotation E(s,,)expressesthesetwhichcontainsthesamepointsasE(S,,)butwithoutcounting multiplicities.WedenotebyEck(SIf,thecountingfunctionofthezerosof卜一awithmulti-

    plicities?kandcountingmultiplicities,Ek)(S,,)thecountingfunctionofthezerosof,一口

    withmultiplicitieskandcountingmultiplicities.ThenotationsE(k(S,,)and)(S,,) expressthesetswhichcontainthesamepointsasEk)(S,,)andEk)(S,,)respectively,but ignoringmultiplicities.

    In1976,Gross[3]posedthefollowingquestion.

    QuestionADoesthereexistafinitesetSsuchthatl|oranypairo|nonconstantentire

functions,andg,E(S,.)=E(S,g)implies,97

    Ifsuchafinitesetexists.anaturalquestionisthefollowing. QuestionBWhatisthesmallestcardinality,orsuchafiniteset? yi[4]firstgaveanaffirmativeanswertoQuestionA.Sofar,thebestanswertoQuestion BwasobtainedbyYi[,asfollows.

    TheoremAThereexistsasetSwith7elementssuchthatE(S,,)=E(S,g)implies {gIoIranypairofnonconstantentirefunctions{andg.

    Later,manyauthorsstudiedthesequestionsformeromorphicfunctions.Thepresentbest ansrertoQuestionBformeromorphicfunctionswasobtainedbyFrankandReinders[. Theyprovedthefollowingresult.

    TheoremBThereexistsasetSwith11elementssuchthatE(S,,)=E(S,g)implies {gforanypairofnonconstantmeromorphicfunctions|andg.

    Anaturalproblemarises:Whatcanwesayifnonconstantmeromorphicfunctions{and ghave''few"poles?In1999.Fangeta1.[71,81provedthefollowingtheorem.

    TheoremCLetS={:76=1).Supposethat,,garetwononconstantmero-

    m.rphicc渤伽s.ti~fyinge(oo,,)>11,

    9(?,9)>11.

    IfE(S,,)=E()

    E(..,,)=E(?,9),then,g.

    Recently,ZhangandXu[9lprovedthefollowingresults.

    TheoremDLetS:{z:.=1).Supposethat,,garetwononconstantmero'

    morphicfunctionssatisfying9(?,,)+0(oo,9)>1.IfE(S,,):E(S,g)andE(oo,,)= E(oo,9),then,g.

    TheoremELetS:{:.=

    meromorphicfunctionssatisfyinge(?,,)+e(.o,g)>

    that{Igaretwononconstant

    4

    .

    IfE(S,,)=E(9)nd

    NO.4QIJ.M.eta1.UNIQUENESSOFMEROMORPHICFUNCTIONSSHARINGONE

SET355

    (?,f)=(?,9),thenfg.

    Inthisarticle,weprovethefollowingresults,whichimprovetheaboveresults

    Theorem1.1LetS={:=1),andkbeapositiveintegersatisfyingk>13. .sen,,.tWOnoneonstantmem.cnctionssatinge(c~,,)>,

    0(oo,9)>.IfEk)(s,),,9.

    ByTheorem1.1,weimmediatelyobtainthefollowingresults. Corollary1.1LetS={z:.:1).Supposethatf,g0retwononconstant mem.nc.s5.tisfying(?,,)>3,

    (?,9)>3.

    IfE(S,,)=E(S,g),e

    fg.

    2Lemmas

    Inordertoproveourresults,weneedthefollowinglemmas Lemma2.1[1oSupposethatfisanonconstantmeromorphicfunction,andP(f)=

    ?pn,,

    Q(,):?qbifJtt(,.c.pmep.lynomi.,,

    e他口,bjnc.nstans

    k=Oj=o

    andap?0,b口?0.'Then

    (r,)=max{,)+.(1).

    Lemma2.2【驯Letf,gbetwononconstantmeromorphicfunctionssuchthatf6(,1)=

    .(g-1).ffo(oo,,)>1.r(..,9)>,e,9.

    Lemma2.3Letkbeapositiveintegerandk?2.Supposethatf,garetwononconstant meromorphicfunctions.Ek1(1,,)=Ek)(1,9),and2e(c~,f)+2e(?,g)+52(0,f)+

    52(0,9)>5+,then,9DrfgL

    Poo.Define

    =

    fll

fl

    gll

    +2.(2.1)

    InvirtueofEk)(1,f)=Ek)(1,g)(k2),letZ0beasimplezerooff1andg?1.Then,

    nearZ0,wehave

    f(z)1=al(zZO)+a2(z)+,

    g(z)1=bl(zzo)+b2(z一询)+,

    where01(?0),61(?0),a2,b2,arecomplexnumbers.Itfollowsthat

    :

    2a2

    +.(),,a

    

    '

    356CoMM.MATH.RESVoL.26 =+

    2a2

    Zzo

    +o(z.?=+,Jf?1,l一口l0,'

    =一一.Z--Z0-4-f1Z0).=一?fJI{|z?v

    =-4-

    gg1Z0.=一———,——一f,fI,一一 'u,'

    Hence()=0.

    Nextweprovethat0.

    Supposeonthecontrarythat?0.SinceEk)(1,,)=Ek)(1,)(?2),usingNevan-

    linna'Sfirstfundamentaltheoremandthelogarithmicderivativelemma

    ,wehave

?1()=?1,(r,1)?(r,1)?,)+o(1)

    ?N(r,)s(r,,)+s(r,9),(2.2)

    and

    ?(r,)(r,,)+(r,9)+z(r,)+z(r,)+?0(r,)

    +?o(r,)+?(7)++(),(2.3)

    wre?o,)denotesthecountingmnctioncorrespondingtothezerosof,,tnataren.t

    zer0s.ff(f1),andNo(r,1)isdefinedana1.gously.ByNevan1inna'ssecondfundamental

    theorem.wehave

    ,

    ,)<,,)+(r,)+(,)??0(r,)r),)1(2.4)

    ,,

    (r,)+(r,1)No(r,1)州协(2.5)

    (r,)+(r,)<?l(r,)+?(r,)

    ?1)(r,)+,9)+.(1),andcombining(2.2)(2.5),wehave ,

    ,,)+,(r,)+,)+(r')

    +,)一?0(r,)一?0(r,1)r,,))

    =

    2I,)+(r,)+z,)+)

    +2()+z(r,)+(r,1)

    NO.4QlJ.M.eta1.UNIQUENESSOFMEROMORPHICFUNCTIONSSHARINGONES

    ET357

    +5/'(r,g)+r,J+rjg)?

    hen

    (2,,)+(r,)+(r,)+?(r,)+2,(r,)

    +.(r,)+(r,1)++)9)

    [2(1-o(o0,,))+12(0,,)+1]r,,)

    +[2(1一臼(..,g))+l(0,9)+](9)+(r,)+s(,,), (r,,)[62o(oo,f)--20(o0,)52(o,,)(0,9)+2IT(r)+s),(2.6) where

T(r1:max~T(r,),,g1,

    and

    S(r)=D(r)),

    possiblyoutsideasetoffinitemeasure.Similarly,wehave

    ()6-20(0o,,)?20(oo,g)-52(0,f)--52(0,9)+](7^)+s(r).(2.7)

    From(2.6)and(2.7)weget [20(c~,f)+20(oo,g)+52(0,,)+z(%)5IT(r)s(2.8) whichisimpossiblesince 20(o0,,)+20(o0,9)+2(0,,)+2(0,9)>5+?

    Therefore,0.

    From(2.1)wededucethat ,=Ag+B,(2.9)

    whereA,B,CandDarefinitecomplexnumberssatisfying

    ADBC?0.

    ByLemma2.1,

    T(r,,)=T(r,g)+0(1).

    Notingthat

    wehave

    29(o.,,)+29(..,9)+2(0,,)+2(0,9)>5+2,

    supA+B<l一而2,(2.1.)

    where

    A=2,,)+2+(r,),B=()+(r,)+.(n). Nowweconsiderthefollowingthreecases.

    Case1.AC?0.

    CoMM.MATH.RESVoL.26

    ,

    )9n

    (r,)+,A)删删

    ==

29(rI,)+(r,)s(r,,).

    But,thiscontradicts(2.10). Case2.A?0,C=0.

    Inthiscase,

    ,.

    ()),

    andwecanusethereasoningofcase1withusedinplace.ft..btainacontradicti.n.

    andSO

    B=0,

    t

    gA

    J'

    Since

    Ek)(1,f)=Ek)(r,g)(k2), thereexists8pointzosuch8(

    f(zo)=g(z0),

    whichyields

    andthus

    Case3.A=0,C?0.

    Inthiscase,

    AsinCase2,wecanprovethat andthus,

    ThiscompletestheproofofLemma2.3

    19

    II

    AD,,

    NO.4QIJ.M.etal,UNIQUENESSOFMEROMORPHICFUNCTIONSSHARINGONES

    ET359

    3ProofofTheorem1.1

    Let

Since

    weknowthat

    F=,.(,1),

    G=96(91).

    Ek)(S,,)=Ek)(S,),

    Ek)(1,F)=Ek)(1,G). (3.1)

    (3.2)

    ByLemma2.1,wehave T(r,F)=7T(r,,)+0(1),T(r,G)=7T(r,g)+Oil).

    Itiseasytoseethat. ()+(r,)?2()+?()

    3T(r,,)+s(r,,)

    (r,F)+S(r,F).

    Then,weobtain..

    ,F)小熙>_.462(oup

    7(3.3),F)=1一熙s丽一?(

    Similarly,wehave

    2(O,G)4.

    (3.4)

    Since

    1,

    e(oo,F)=1一专(1e(oo,,)),e(oo,G)=1一毒(1e(oo,9)),

    weconcludefromtheassumptionofTheorem1.1that

    e(oo,F)+0(oo,G)>.(3.5) Combining(3.3)(3.5),wehave 2e(oo,F)+20(o0,G)+(0,F)+2(0,G)>5+?(3?6) ThenbyLemma2.3,

    FGorFG1.

NowWeprovethatFG?1.

    Otherwise,weWouldhave

    ,.(,1)g.(91)1.(3.7)

    Since

    )(S,f)=Ek)(S,g)(k>13),

    weknowfrom(3.7)thatthemultiplicityofthezerosof,Cisatleastk+1(wherecis

    oneofthesolutionsoftheequation70=1),andthemultiplicityofthezerosof,1 isatleast7.UsingNevanlinna'sfirstandsecondfundamentaltheorems,weobtain

    )+(r,)+(r,,)+s(r,,) (r,,),

    COMM.MATH.R.ESVoL.26

    ?南(r,,)+(r1,)+(7.,,)+s(r,,).

    Accordingtotheassumption

    I,)>(>l3),

    andcombining(3.8),weseethat (1一南一毒)(rI,)()?k-13().

    Obviously,(3.9)cannothold. Sowehave

    thatis,

    FG.

    ,.(,1)=g6(夕一1).

    Accordingtotheassumption,weseethateither

    I,)>.rp()>

    ByLemma2.2,wehave.

    {g.

    TheproofofTheorem1.1iscompleted. Rferences

    (3.8)

    (3.9)

    [11Hayman,W.K.,MeromorphicFunctions,ClarendonPress,Oxford,1964.

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