DOC

Uniqueness of Entire Functions Concerning Differential Polynomials

By Debbie Cooper,2014-02-19 06:02
8 views 0
Uniqueness of Entire Functions Concerning Differential Polynomialsof,OF

    Uniqueness of Entire Functions Concerning

    Differential Polynomials

    Chin.Quart.3.o1Math

    2010,25(2):214219

    UniquenessofEntireFunctionsConcerning

    DifferentialPolynomials

    XIONGWeiling

    (DepartmentofInformationandComputingScience,GuangxiInstituteofTechnology,Liuzhou545006,

    China)

    Abstract:Inthispaper,wedealwiththeuniquenessproblemsonentirefunctionscon- cerningdifierentialpolynomialsthatshareonesmallfunction.Moreover,weimprovesome formerresultsofMFangandWLin.

    Keywords:uniqueness;meromorphicfunction;entirefunction;sharevalue 2000MRSubjectClassification:30D35

    CLCnuYnber:O174.52Documentcode:A

    ArticleID:1002?0462(2010)02-0214-06

    ?1.IntroductionandResults

    Inthispaper,weassumeallthefunctionsarenonconstantmeromorphicfunctionsinthe complexplaneC.

    WeshallUSethestandardnotationsofNevanlinnatheoryofmeromorphicfunctionssuchas T(r,,),m(r,,),N(r,,),g(r,,),S(r,,),etc.

    SetE(Q,f)=z}zerosoff(z)?()withmultiplicitymtimes}.IfE(,f):E(a,),

    thenwesaythatf(z)andg(z)share(z)CM.SetEk)(n,f)=zfzerosoff(z)Q()

    withmultiplicitym).Obviously,ifE(,f)=E(,9),thenEk)(d,f)=Ek)(a,9),for =1,2,?.

    Let,beameromorphicfunction.Wedenotebynk)(r,f)thenumberofpoleswithmulti

    plicityatmostkof,inIZl<rcountingitsmultiplicities.Wedenotebyn(k(r,f)thenumber ofpoleswithmultiplicityatleastkoffin<rcountingitsmultiplicities.Wedenoteby n2(r,,)thenumberofpolesof,in<r,whereasimplepoleiscountedonceandamultiple Receiveddate:2006.O904

    Foundationitem:SupportedbyNSFofGuangxiProVince(0728O41)

    Biography:XIONGWeiling(1959

    ),female,nativeofLiuhzhou,Guangxi,aprofessorofGuangxiInstitute ofTechnology,engagesincomplesanalysis.

    No.2XIONGWeiling:UniquenessofEntireFunctionsConcerning215

    poleiscountedtwotimes.Wedenoteby(r,f)asthecountingfunctionof(r,,)counted

    withignoringmltiplicities.

    (r,f)andN(r,f)aredefinedinthetermsofn(r,f)andr,f)intheusualway,

    respectively?Inthesameway,wecandefineNk)(r,JL?

    ),N(k(r,)and?2(r,)andsoon.

    ItiswellknownthatiffandgsharefourdistinctvaluesCMthen{isaMSbiustransformation ofg.Recently,correspondingtoonefamousquestionofHayman[,manyuniquenesstheorems

    forsomecertaintypesofdifferentialpolynomialssharingonevalueweobtained(see[2?5]).

    In2001,MFangandHongproved:

    TheoremA【引Letfandgbetwotranscendentalentirefunctions,n11aninteger.If ,"(,1)fandgn(9lbshare1CM,then,()g(z).

    Afterwards.WLinandHYiimprovedTheoremAobtainedthefollowingresult TheoremB【】Let,andgbetwotranscendentalentirefunctions,7aninteger.If

    .(,1)fandgn(glbshare1CM,then,(z)9(2).

    Inthispaper,someuniquenessquestionsofmeromorphicfunctionsareinvestigated,which areimprovementandcomplementaryfortheaboveresult.

    Theorem1Let,andgbetwotranscendentalentirefunctions,Qbenonzeromero

    morphicfunctions,mandpositiveintegers,T(r,)=o(T(r,,)),T(T,Q)=o((r,9)).If )(,,n(,?1)mf)=Ek)(a,gn(夕一1)mg)andn>max{,3rn+2)aninteger,then

,g.

    Remark1UndertheconditionofTheorem1,letk-?oo,wegetthattheresultof TheoremlisstillvalidifE(,f(,1).)=(0,gn(1)g)and>max{6,3m+2). Whenm=1,ifn>6,wehave,g.Whenm:2,ifn>8,wehave,g.Whenm=3,if n>11,wehave,9.Obviously,Theorem1improvesTheoremAandTheoremB.

    ?2.Lemmas

    Lemma1[.Let,()beanonconstantmeromorphicfunctionand nm

    R(,)=?akf/?ajr

    k=0j=0

    beanirreduciblerationalfunctioninfwithconstantcoefficients{Q&}and{6j),wherean

    ?0

    andb?0.ThenT(r,R(,))=dT(r,,)+S(r,,),whered=max{n,m}. Lemma2Let,and9betwononconstantmeromorphicfunctions,nandmbepositive

    integer,T(r,oL)=o(T(r,,)),T(r,Q):o(T(r,9))and&?0,oo,andlet

    F=,"(,1)f,G=9n(91)g.

    IfEk)(Q,F)=Ek)(ol,G)and(n6)km4,thenS(r,,):S(r,g)

    ProofbyLemma1,wehave

    (+m)T(r,,)=T(r,f(,1))+S(r,f)T(r,F)+T(r,,)+S(r,,) 216CHINESEQUARTERLYJOURNALOFMATHEMATICSVb1.25 Therefore

    T(v,F)?(礼十m2)T(r,f)+S(r'f)

    Bythesecondfundamentaltheory,wehave

    T(r,F)N(r,F)+,

    

    N(r,F)+N(r,

    

    N(r,F)+Nr,

    (4+m)(r,,)+1(r

    ,F)+(r,G)+s(r,,).

NotingthatT(r,G)T(r,g(夕一1))+(r,g)(+m2),9)+S,9),wededucethat

    f?_-'...'--'.........'

    ,(+m

    +1

    ThusS(v,,)=s(v,g).

    Lemma3f5

    Ek)(1,G),andlet

    

    4m)T(r,f)(n+m+2)T(r,g)+S(r,)+S(r,9). LetFandGbetwononconstantmeromorphicfunctionssuchthatEk)(1,F)=

    =

    c?2H2

    IfH?0,then

    {(r,F)+(r,G))?2(r,F)+?2(r,1)+?2(r,G)+?2(r,1)

    +((r,)+((r,)+(r),

    whereTff)=max{T(r,F),T(r,G)},s(r)=D((r))(r..,7'E),Eisasetoffinitelinear

    mea$Ure.

    T.pmma4[7]LetbedefinedaSinLemma3.IfE(1,F)=E(1,G),H0and lim

    -p?I

    !:圭?!:查?!!!?翌::2T(r)<1,

    whereIisasetwithinfinitelinearmeasure,thenFG1orFG

    Lemma5Letfandgbetwotranscendentalentirefunctions,Qbenonzeromeromorphic

    functions,mand%positiveintegers,T(r,0):.(,,)),T(r,Q)=.((r,9)).IfEk)(a,,(, 1)m,)=)(,g(夕一1)g),thenwhen(七一1)n>6+2m+10,H0Where ProofLet

    where

    1)mg 9"(9

    Fl=,"+,

    ,

:(2F;I)_(GII

    2

    ).

    9n+lG.

    :

    m

    _

    jm-j

     ,mJ,gm'

    (2.1)

    No.2XIONGWeiling:UniquenessofEntireFunctionsConcerning

    ByLemma1andLemma2,wehaves(r,,)=s(r,9)(=s(r)say)and

    217

    T(r,F1):(n+m+1)T(r,,)+S(r,,),T(r,G1):(n+m+1)T(r,g)+S(r,9).(2.2)

    Since=aF,wededuce

    (r,F1)

    (r,F)+?(r】击)J7v(r,1)+s(r) :(r,F)+(n+1)?(r,71)+?(r,

    )n?(r,71)roW()一?(r,1)+s(r) =

    (rF)+?(r,)+?(r,)m?(r,)一?(r,1)+s(r). Inthesamemannerasabove,wehave Thus

    (G)(r.G)+?(r,1)+?(r

    ,

    )raN(r,1)一?(r,1)+s(r)

    ,)+,)}-m{?(r,)+?(r,)}

    If日?0,byLemma3,wehave {,F)+(r,G))

    (r,F)+?2(r,)+?2(r,G 271)+,))+?2

    )+?2(r,)++1(r, )+?2(r,

    +r,

    )+r,c1--~_1)+) From(2.2),(2.3)and(2.4)wehave

    (n+,n+1){(r,,)+(r,9))

    By(2.2)wehave 1

    

    1

    )+

    +S(r).(2.3) )+s(r)

    J

    (n+m+1){T(,,,,)+(r,9)){7+m+){T,,)+T,9))+s(r)

    Thus+m+17+m+,whichc.ntradicts(七一1)>6k+2m+10

    (2.4)

    (2.5)

    ?

    +

    

    ?

    +

    1g

    ?

    +

    1,J

?

    _}.

    ++

    <

    

    ?

    

    ,l,

    一广

    ?

    

    l

    +

    I

    

    _l

    ,

    ?

    .

    r+

    ,

    <

    218CHINESEQUARTERLYJOURNALOFMATHEMATICSVl01.25

    Lemma6【】Let,andgbetwoentirefunctions.Supposethatthereexisttwononcon-

    stantpolynomialsPandqsuchthatPof(z)=q.9().Thenthereexistanentirefunctionh

    andrationalfunctionsU(z)andy(z)suchthat

    f(z)=Uo(z),g(z)=Voh(z) ?3.ProofofTheorem

    ByLemma5,wehaveH0.Thatis2atl2.Hence1+B,

whereA?0andBareconstants.ThusE(1,F)=E(1,G),andT(r,F)=T(r,G)+S(r,,).

    Since

    thuslim

    (r,F)+(r,1)+(r

    ,

    G)+(r,1)

    (r,,)+(r,)+(1)+(r ,f

    l

    --

    ~_1)

    +(r,)+(r,)+(r,1)+(r ,

    1

    g)+S(r)+?(r,)+?(r,)+?(r,)+?(r,———)+) 00'口?I

    5{(:,,)+(r,9))+s(r) (r)+s(r),

    !!!2?翌!!!2?翌!!!!)?翌!!!2 Tir1

    r?,

    thefollowingtwocases <1.byLemma4wegetFG1orFG.Wediscuss

    Case1SupposethatFG1.thatis ,"(,1)g(9?1)gQ

    Since,andgentire,by(9),wehaveN(r,)+N(r,,)+N(r,1):'5-(,).Weget

    constantfunction,contradicts.

    1)fgn1)g.Thus Case2SupposethatFG,then"(f?

    Where

G1+C

    R:,+,Gl=gn+lG2,F2=.,,G.= j=o

    ~

    (77)7i

    ByLemma2haveS(r,f)=S(f.,9)(:s(r)say).ByLemma1have

    Thus

    T(r,)=(+m+1)T(r,,)+S(r),T(r,GI)=(n+m+1)T(r,g)+s(r)

    T(r,)=T(r,g)+S(r).

    (3.1)

    {isa

    (3.2)

    (3.3)

    "

    .J

    ?

    No.2XIONGWeiling:UniquenessofEntireFunctio

    nsConcerning219

    SupposethatC?0.By(3.3)wehave (n+m+1)T(r,g)=T(r,G1) <N(r,)+N(r,L1G1+C)+N(r,G1)+S(r)

    ,

    )+,)+71)+,1)r)

    ?(2m+2)T(r,g)+s),

    thusn+m+12m+2

    ,

    whichcontradicts>3m+2.

    ThereforeGl,thatis

    ,卅?

    j=0

ByLemma6,thereexistrationalfunctionsUandV

    ,andentirefunction^,suchthat Thus

    f=(),g=(^).

    (1)UV"('/1)V

    "+l

    j=o

    ~

    (;U)jTiUj=_vn+1

    J:.,

    (

    .

    --

    1'JCJm

    (3.4)

    (3.5)

    (3.6)

    (3.7)

    Since/andgbeentirefunction,thusUandVbepolynomialsoronlyhave0nepo1e.

    Since?.?0,by(3.6)and(3.7),wehavecandshare0and?cM.Thus UkV,whereconstant.

    By(3.6),wehave:1.HencejF_9.

    ThiscompletestheproofofTheorem1 [References]

    []HAYMANWK.ResearchProblemsinFunctionTheory[M].London:AthlonePress.1967

    .

    YANGChung'chun,XINHouhua.UniquenessandvalueshareingofmeromorphicfunctjonsJAnnAcad SciFennMath,1997,22:395406.

Report this document

For any questions or suggestions please email
cust-service@docsford.com