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Value Distribution of a Class of Differential Polynomials

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Value Distribution of a Class of Differential Polynomialsof,a,A,Value,Class,value,class,CLASS

    Value Distribution of a Class of Differential

    Polynomials

    Chin.Quart.J.o|Math

    2010,25(1):23?29

    DifferentialPolynomials

    XUJun-feng,ZHANGZhanliant~

    (1_DepartmentofMathematicsandPhysics,WuyiUniversity,Jiangmen529020,China2.Depnrment

    ofMathematics,Zhaoqin9University,Zhaoqing526061,China)

    Abstract:Letf(z)beatranscendentalmeromorphicfunctioninthecomplexDlaneand a?0beaconstant,foranypositiveintegerm,,k,satisfymnk+n+2,=,+0(,())"

    hasinfinitelymanyzeros.Thecorrespondingnormalcriterionalsoisproved. Keywords:meromorphicfunction;differentialpolynomials;zeros;normalfamilv 2000MRSubjectClassication:3OD35

    CLCnumber:O174.52Documentcode:A

    ArticleID:10020462(2010)01002307

    ?1.IntroductionandMainResult

    In1959,WKHayman[Jfirstconsideredandinvestigatedtwoclassesvaluedistributionsof }i}nand}|af"a?0isaconstant1.Forthevaluedistributionsofftfnthoughtthework ofHayman,EMues[,

    WBergweiler[7.andSOon,finallyWaSsettledbyHuaihuiChen6].For

    theclassfunctionof,afa?0isaconsent),by[6]weknown3,theconjectureof

    Haymanistrue.Meanwhile}Ye[hasprovedsomeinterestingresultsas|+}.Inthispaper, wediscussmoregenerallycase:

    Theorem1.1Letfbetranscendentalmeromorphicinthecomplexplane.Leta0be aconstant,=,+0(,())",foranypositiveintegerm,n,,satisfymnkn+2,

    Thus,weimmediatelyget

>0

    Recelveddate:20061208

    Foundationitem:SupportedbytheNSFofchina(1O77l121);Supportedbythe''Yumiao''Proj

    ectof

    GuangdongProvince(LYM08097)

    Biographies:XUJun~ng(1979

    ),male,nativeofNanzhang,Hubei,Ph.D.,engagesincomplexanalysis; ZHANGZhanliang(1962

    ),male,nativeofHezhe,Shandong,aprofessorofZhaoqingUniversity,engagesin complexanalysis.

    24CHINESEQUARTERLYJOURNALOFMATHEMATICSVl01.25

    Corollary1.1Ifforanypositiveintegerm,n,k,satisfymnk+n+2,= ,+0(,())"

    hasinfinitelymanyzeros.

    In[6]jChenandFanghaveposedthefollowingconjecture ConjectureLet,betranscendental

    positiveintegerandm,satisfym+2,

    meromorphic,anda?0beaconstant,forany

    then,()afhasinfinitelymanyzeros.

    RemarkInTheorem1.1,letn:1,thentheconjectureholdsatm+3.Ingenerally,

    ifm=k+2,theconjectureistrueunlessfhastheadditionalconditions.Forexample:Xu

    ExampleLetf=1/(1+e.)andF=,f..Thenf=3f(f1)andsoF=

    

    (,1).1hasnozerobecausefdoesnotassumethevalue0.Theexampleshowsthe conjectureisn'ttrueiffisnotdefinedinm=k+2.

    Theorem1.2LetbeafamilyofmeromorphicfunctionsonadomainD, ifallzeros

    0,a?0,forany offunctionsinhavemultiplicityatleastk,andsatisfyf+0(,())?

    positiveintegerm,n,,ifmnk+Tt+2,isnormalonadomainD.

?2.SomeLemmas

    Lemma2?1Letfbemeromorphicinthecomplexplaneand0?0beaconstant,for anypositiveintegerm,n,,satisfymnk+n+2,iff+n(,())"?0,thenfisthe

    constant.

    ProofIffisnottheconstant,byTheoremI.1weknow,isnottranscendentM,andif

    ,istherational,wecanexaminef+0(,()musthavethezeros,itisac0ntradiction.

    Lemma2.2Letfbemeromorphicinthecomplexplane,anda?0beaconstant,for anypositiveintegerm,,k,satisfynk++2,iff+0(,())0,thenfisaconstant. ProofIf1isnottheconstantbytheconditionweknowlisentireOtherwiseifz0is

    thepoleofPorderoff,thenmp=+),contradictedwithmnk+n+2.Thenwith theidentityf三一.(,(')"or(,)=--(),wecangetthatifr}?,

    1f1

    m一咒)(r,,)=(m一佗)m(r,,)log++nm(r,)=0((r,)),

    f?IJ

    and7'EwithEbeingasetofrvaluesoffinitelinearmeasure. ?3.ProofofTheorem1.1

    Theproofisindirect.Withoutlossofgenerality, leta=1,

    =0

    p

    S

    n?

    No.1XUJunfengetal:ValueDistributionofaClassof25 SinceT(r,)=O(T(r,,)),weget

    Bydifferentiatingtheequation

    /m

    ,)

    =/+(,()

    1&

    =

()-g

    S(r,/)

    )",wehave

    

    (,())n--i,(+)

    where,_,and

    =(m/,.)

    ApplyingClunieLemmato(3),wehave m(r,a)=S(r,/)

    (1)

    (2)

    ThepolesofQcanonlyoccuratthepolesof/andthezerosof.With(2),thepolesoffare

    notthepolesofQ.By(1),weget

    Hence

    ?()SN(r,)=s()

    T(r,Q)=S(r,/)

    If0,thenm?=.Byintegrati.n,

    Notethatmnk+n+2,itisimpossible, again,weget

    Hence

    then

    e/=,c?0isaconstant,or(cI)/=,(.

    hence?0.NextapplyingClunieLemmato(2) m(r,,):s(r,.).

    m(r,,)=m(,.)m(.,)+m(r,)+o0)m(r,,)+(r,)+D(1), m(r,/)=s(r,,)(5)

    With(2)?(3),weknowthepolesofqorderof,atleastisthezerosof(m1)q[n(+)+1]= (m一礼一1)q(nk+1)orderof,hence,ifm>nk+n+2,then

    ?()?(r,)+,,)7

    thenwith(5),wecangetthecontradiction.Thus,wehavem:nk+n+2,and

NI(r,,)?(r,1)+s(r

    ,,).

    Nowwith(2)and(3),eliminatingthetermscontaining,andnotingm=nk++2,then have

    5(/++.(,()))=[(nk+n+2)f(,())n/f(+'(,(')一】

    26CHINESEQUARTERLYJOURNALOFMATHEMATICSVol_25

    Let0isthesimplepoleoffandnotthezeroofOl,theninaneighborhoodofZ0,

    f(z):

    ZO

    +d0+O(z)(dl?0),

    a(z)=(z0)+O/(0)(zo)+0((zo).)((0)?0)

    puttheminto(7)andthoughtthecalculating,weget

    d+(1)+2k!

    2

    (+1)+2(-1)"+(n+k+2)

    InthealgebraicfunctionofW=(z)determinedbytheequationofa(z)w+:(1)+2k!, takingabranchofw(zo)=dl,wedenotebydl(),thenlet

    do(Z)=2

    (+1)+2(1)+(nk+k+2)

    thendl(Z0)=dl,do(Z0)=do,SO

    With(4)

    f(z)=d1(0)

    Z——ZO

    d(),

    a(zo,,'

    +dozo)+O(zzo)

    T(r,d1)+T(r,do)=S(r,f)

    Thoughtthesimplecalculating,weknow ):八卅1(2d0()

    isanalyticatZ0,with(5)(7),wegetT(r,Ao)=S(T,.).Thusiflet =

    (2d0,.=1

    thenfsatisfiesRiccatiequation whereA2(z)?0and

    By

    W=A0()+A1(z)+A2(),

    f=Ao+Alf+A2f.

    Using(12)overandoveragainwededucethat

    ,()=k!Ak(z)f++,

    (8)

    (10)

    (12)

    d

    ))

    

    (,)

    =

    A

    .

    ?=

    No.1XUJunfengetal:ValueDistributionofaClassof27

    arepolynomialsinf.ThuswedenotetherightofthepolynomialsbyG(z,,),furthermOre,

    isalsopolynomialsinf.

    =

    ,n+n+2+(.())n=fnk+n+2+Gnz,

    WedefinethefunctionofZ,W H(z,):W+++G(z,W)(13)

    ThisisapolynomialinWwhichsatisfiestheidentityH(z,,(z)).Obviously,ifW:is

    thesolutionofRiccatiequation(10),thentheremustbe(,/(z))三一rlnk+n+.(z)+ri(k)(z). Let

    H(z,W)=WR(z,

    whereR(z,)ispolynomialinWandR(z,O)?0.Considering

    R(z,W)=0

    (14)

    ItssolutionWri(z)isalgebraicfunctionandsatisfiesT(r,ri):s(r,,)(cf[10).Inthe

    following,wefirstprovethesolutionof(15)satisfiesRiccatiequation(10),and

    Werewrite(2)intheform

    

    W+..+(())"=0

    (,n+n+Ol十仃(,())扎一,(+),()=0

    NotethatH(z,,),(.,())a(z,,)and,satisfiesRiccatiequation(10),thatis Or

    Q,"+n+(z,,)+(G(,,)+G(z,f)f)H(,,)G(z,,)((,,)+;(z,f)f)0

    (,+++G(z,,))(,,)?c(z,,)(z,,)+(G(z,f)H(z,')

    

    C(z,.)(z,,))(A0+Alf+A2f.)0.

    Therightoftheaboveequationisthepolynomialinf,andmustbeidenticallyzero.

    Thatis

    (Q"+"++G(z,))(,)C(z,)(2,)+(G(,)(,)

    

    C(z,))日乞(z,w)(Ao(z)+A1(z)w+A2(z)fw.)0

    forarbitrarycomplexandW.

    Let=()bethesolutionofthefunctionequation(15).If C(z,ri)0,thenwith(13),weknow770,itcontradictswithn(z,0)?0.HenceC(z,)?0.

    SinceWri(z)alsoisthesolutionofthefunctionequationH(z,)=0,thereisaunique

    positiveintegersuchthat

    H(z,W)=(W77(2))H(,),H(,)?0

    Bycombiningitinto(17),weknowW=(z)satisfiesRiccatiequation(10).Thefunction equation(15)atleasthasonesolution(multiplyvaluedormeromorphic).Ifitonlyhasone CHINESEQUARTERLYJOURNALOFMATHEMATICSV01.25 solution:(),then,)=()+"+._..Since,)hasnotthenexthighterm,

    SOtheremustbe:0,itcontradictstheabovediscussion.Ifithasatleastthreedistrict

    solutionsl,772,

    3,thentheRiccatiequation(10)atleasthasfourdistrictsolutions,rll,772,r13,

    becausethecrossratioofarbitraryfoursolutionsofRicaatiequationisaconstant(cf.[111),

    thenIcanbedenotedbytherationalfunctionsof7/1,r12,3,then

    2

    (r,,)=0(?(r,))=s(rj,)j

    j=o

    itisimpossible.Thatis,thefunctionequation(12)justhastwodistrictsolutions(multiply

    valued),then

    R(z,W):(W一叼l())(W一叩2(z)).,

    whereA1,2arethepositiveintegerand1+2=nk+n+2z.SinceR(z,W)hasnotthe nexthighterm,SOA1T]I+A2r/20,thisis1=P'?72,P=.

    Sincer/i,2satisfythedifferentialequation(13)whichyields

    口叼;+n++()n0

    

    0(p2)n+2p(5)n0

    thenP+(-p)"++.=0(2?0),henceP:lor1

    IfP1,wehave1=2,l=-r12,and

    H(z,W)=W(.一叼.)

    forand一叼arethesolutionsofRiccatiequation(10),then With(9),wehave

    whereM:

    Byintegration

0+A27720,

    ,,——

    IVI1'——

    "

    C1QM,

    

    :A

    

    (19)

    whereC1?0isaconstant,nextwith(13)and(18),weget!"5=一叩,therefore,by(9), =c2Q

    where?0isaconstant,where=

    constantfunction,thuswith(19)and(2o), (20)

    nk.

    For77satisfies(16),weknowisnota

    wecangetacontradiction.

    IfP:1,wecangetacontradictionsimilarly.ThuswecompletetheproofofTheorem

    1.1.

    NO.1XUJunfengetal:ValueDistributionofaClassof29 ?4.ProofofTheorem1.2

    ProofWemayassumethatD:?.Supposethatisnotnormalon?.Then.taking =

    nk

    ,then0<OL<k,andapplyingZaclmanLemma[tog={1/f:,?F),we canfind乃?F(j=1,2,'?'),zj-?z0and(>o)-?0suchthatg5(0:p~ij(zj+(), convergeslocallyuniformlywithrespecttothesphericalmetricto((),wheregisanonconstant

    meromorphicfunctiononC.ByLemma2.1,thereexists(b?{IzIR)suchthat .q((O)+0(9()((0))"=0

Fromtheaboveequality_9((b)?...Throughthecacualation

    9(()+.(9(())=p((()+.((())n)?0

    Theleftofequalityconvergesuniformlytogm(<)+n(.g()(()),byHurwitztheorem,weknow

    theequalityisidentityzerooridentitynonzero.From(21),weknow9TM(()+0(_9()(())"

    0,

    thenbyLemma2.2yields9(()isaconstant,itisacontradiction.Hencewecompletetheproof ofTheorem1.2.

    [References]

    f1]HAYMANWK.MeromorphicFunctions{M].Ox[ord:ClarendonPress,1964. [2]HAYMANWK.Pieardvaluesofmeromorphicfunctionsandtheirderivatives[J].AnnofMath,1959,70:

    942.

    3]MUESE.UbereinproblemyonHayman[J].MathZ,1979,164:239259.

    [4]YEYa-sheng.Apicardtypetheoremandblocklaw[J].ChinMathAnn(B),1994,15:75

    80.

    5]XUYan.Normalfamiliesofmeromorphicfunctions[J].JournalofMath,2001,2l(4):381385.

    6]CHENHuaihui,FANGMingliang.Onthevaluedistributionof,,,[J

    _SciChinaSetA,1995,38:789798.

    [7

    BERGEILERW,EREMENKOA.Onthesingularitiesoftheinversetoameromorphicfunctionoffinite

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    8]CHENHuaihui,GUYong

    xin.ImprovementofMarty'Scriterionanditsapplication[J].SicenceinChina, SeriesA,1993,36(6):674681.

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