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# Uniqueness of Entire Functions Concerning Differential Polynomials

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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

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&gt;max{,3rn+2)aninteger,then

,g.

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

?2.Lemmas

Lemma1[.Let,()beanonconstantmeromorphicfunctionand nm

R(,)=?akf/?ajr

k=0j=0

beanirreduciblerationalfunctioninfwithconstantcoefficients{Q&amp;}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&amp;?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)&lt;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&gt;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)

(2.4)

(2.5)

?

+

?

+

1g

?

+

1,J

?

_}.

++

&lt;

?

,l,

一广

?

l

+

I

_l

,

?

.

r+

,

&lt;

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 &lt;1.byLemma4wegetFG1orFG.Wediscuss

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

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

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) &lt;N(r,)+N(r,L1G1+C)+N(r,G1)+S(r)

,

)+,)+71)+,1)r)

?(2m+2)T(r,g)+s),

thusn+m+12m+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

.