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SOME POLYNOMIAL INEQUALITIES IN THE COMPLEX DOMAIN

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SOME POLYNOMIAL INEQUALITIES IN THE COMPLEX DOMAIN

    SOME POLYNOMIAL INEQUALITIES

    IN THE COMPLEX DOMAIN Ana1.TheoryApp1.

    Vo1.26,No.1(2010),1-6

    DOl10.1007/s1049601000017

    SoMEPoLYNoMIALINEQUALITIESIN

    THECoMPLEXDoMAIN

    K.K.DewanandS.HanS

    (CentralUniversity,IndiaJ

    ReceivedDec.20,2008

    ?EditorialBoardofAnalysisinTheory&ApplicationsandSpringer

    VerlagBerlinHeidelberg2010

    Abstract.Let

    P(z)

    beapolynomialofdegree,zandletM(f,r)=maxif(z)fforanarbitraryentirefunction

    Zl=r

    (z).IfP(z)hasnozerosinlzI<1with(,1)=1,thenforlI1,itisprovedby Jain[GlasnikMatemati~ki,32(52)(1997),4551]that

    +

    ()!)f

    )lR1,Izl1

    Inthispaper,weshallfirstobtainaresultconcerningminimummodulusofpolynomials

    andnextimprovetheaboveinequalityforpolynomi~swithrestrictedzeros.Ourresult

    improvesthewellknowninequalityduetoAnkenyandRivlin[1andbesidesgeneralizes somewellknownpolynomialinequalitiesprovedbyAzizandDawood[31. Keywords:polynomial,inequality,restrictedzeros AMS(2010)subjectclassification:30A10,30C10,30E15

1IntroductionandStatementofResults

    IfP(z)isapolynomialofdegree,then[,p?,pr.m,p?

    M(P,R)R1)forR1

    Theresultisbestpossibleandtheequalityholdsforpolynomialshavingzerosattheorigin

    (1.1)

    SupportedbyCouncilofScientificandindustrialResearch,NewDelhi,undergrantF.No.9/4

    66(95)/2007EMRI

    ?

    +2+2RR/,?,\/,?,\

    ++

    l

    12+

    2K.K.Dewanetal:SomePolynomialInequalitiesintheComplexDomain Inequality(1.1)wasgeneralizedbyJain[41whoprovedthatifP(z)isapolynomialofdegree

    n,thenforIZJ=1and10fJ1,

    ()"?().2,

    ItwasshownbyAnkenyandRivlin[thatife(z)?0inIZI<1,then(1.1)canbereplaced by

    Rn

    z

    +l

    M(P,1)f0r1.(1.3)

    Inequality(1?ljJISsharpandtheequalityholdsforPiz)+nn,whereIJ=lyJ=l/2.

    Inthesamemannertheinequality(1.3)wasgeneralizedbyJain[forpolynomialshaving

    nozerosinIZI<1withIl1andIZI=1,

    ()nP(z)I

    +

    ()I+JRnq-ot().4

    Inthispaper,wefirstlyobtainaninterestingresultconcemingminimummodulusofpoly

    nomialsP(z)whichisanalogoustotheinequality(1.2).

Theorem1.IfP(z)isapolynomialofdegreen,havingallitszerosinlZI<1,thenfor

    everyrealorcomplexnumberwithII1andR1,

    min

    i+()n)1->Rn+a(JzJ=1min)I.(1?5)

    TheresultisbestpossibleandtheequalityholdsforP(z)=me,>0.

    Ifwetake=0inTheorem1,thentheinequality(1.5)reducestothefollowingresult

    provedbyAzizandDawood[.

    Corollary1.LetP(Z)beapolynomialofdegreehavingallitszeros<1,thenfo

    ZI:1

    P(Rz)l>Rnmilzl:

    n

    1

    IP(z)I.,.1

    Wenextimprovetheinequality(1.4),byusingTheorem1.Moreprecisely,weprovethe

    following

    Theorem2.P(Z)isapolynomialofdegreen,havingnozerosin<1,thenfo,_every

    realorcomplexnumberwithIf1,R1andIzI=1,

    })P(z)

    fR+1【丁

    I+I1+()

    mi

    lzl:

    n

    l

    iP(z)

    (1.6)

    r,

    ,/

    

    /.

    .

    ,'

    1-

    +

    f.,

    ===

    r??????L

    l2

    <

    ?一

    +2+尺一一.,.rJ,L十一

    R

    P

    AnaJ.TheoryApp1.,Vo1.26,No.101D)3 Inequality(1.6)issharpandtheequalityholdsfor(z)=+yz,whereJI=JI:1/2. For=0,Theorem2reducestothefollowingresultprovedbyAzizandDawood[3].

    Corollary2?Lete(z)ispolynomialofdegree,z,havingnozerosinzf<1,thenfor lzJ=1

    (Rn;1)M(P,-,?()

    2Lemmas

    Fortheproofofthesetheorems,weneedthefollowinglemmas.

    Lemma1.ifP(z)isapolynomialofdegreen,havingallitszerosinthediskk1,

    thenforR1

    lP(I(R+k)z)IIzI:1.(2.1)

    TheaboveLemmaisduetoAziz[2].

    Lemma2.LetF(z)bepolynomialofdegree,havingallitszerosinthediskIZ11.If

    P(z)isapolynomialofdegreeatmostnsuchthat thenforlf1andR1,

(z)I{(.)Jfo,.f=1,

    +

    ()P(z)1()-z-2

    .fheaboveLemmaisduetoJaint"1.

    Lemma3.1fp(z)isapolynomialofdegreen,thenrlzIZ=1andlf1,

    ()+()Q(z)J

    JRn+a()+().3

    whereQ(z)=z.

    ThPbo,,PT,mmid?ptnlain[4

    3ProofofTheorems

    ProofofTheorem1.ForR=1theresultisobvious.Thereforeweshallprovetheresult forR>1.IfP(z)hasazeroonIzl=1,thentheinequality(1.5)istrivia1.Sowesuppose thatP(z)hasallitszerosinIzl<1.Ifm:rainIzf=1IP(z)l,then0<IP(z)Jfor=1. 4K.K.Dewanetal:SomePolynomialInequalitiesintheComplexDom~n Therefore,ifisacomplexnumbersuchthatl

    f<1,thenitfollowsbyRoucheSlheoremmat

    thepolynomialG(z)=P(z),,zz,lofdegreen,hasallitszerosinlzI<1.ApplyingLemma1 tOthepolynomiala(z1withk=1andR>l,weget

    IG(Rz)I(,G(z)If01.

    SinceG(Rz)hasallitszerosinlzll<l,againapplyingRouche'STheoremforrealorcorn

    plexnumberawith[al1,.necansh.wfhatthep.lyn.mia1T(z):G(Rz)+,)nG(z)

    hasallitszerosinIzl<1.Thatis,

    (z):G(z)+(,,)G(z)?of0rlzl>1,>1.

    SubstitutingforG(z),weconcludethatforevery,withIJ<1,II1,lzI1andR>1 =

    +

    ()P(z)]mRnznq-a()"?

    Thisimpliesforeverywithll1,lZI1andR>1,

    ()P(z)l()2

    Ifthisinequalityisnottrue,thenthereisapointZ=Z0withIZ0I1suchthatforR>1

l(Rz()P(z)I<IRn-+-a()

    Wetake

    :

    ?(-z-)P(zo),

    Rn+()l

    thenf<1andwiththischoiceof,wehavefrom(3.1),T(zo)=0forIZ0l1.Butthis

    contradictsthefactthatT(z)?0for1.Henceinparticular,(3.2)givesforeverywith Ij1andR>1,

    ()P(z)IfRn+a()"』?-.

    ThiscompletestheproofofTheorem1.

    ProofofTheorem2.ForR=1thereisnothingtoprove.ThereforeweassumethatR>1.

    Byhypothesis,thepolynomialP(z)?0inlZI<1,thereforeifminfzf:lJp(z)f,then IP(z)IforlzI1.Therefore,foragivencomplexnumberpwithII1,itfollowsbyRouche'S TheoremthatthepolynomialG(Z)=P(z)一卢hasnozeroinZI<1.Nowif

    /4(z)=ZnG(1/-~)=Q(z)JB,

    Ana1.TheoryApp1.,Vo1.26,No.Jr20J0)5

    thenallthezeros(z)lieinzJ<1andIC(z)l==:IF(z)Ifor=1.ThereforebyLemn1a2, we

    haveforIl1and1ZI=l,

    +

    ()

    (Q(()Thisimplies

    ()叫卢{?+())

    ,}J

    ?

    (R2~1)nQ?)_())f.3

    SinceallzerosofQ(z)lieinIzl<1,wehavebyTheorem1for===1andI11 +

    ()?Rn+O~

    ::Rn+o~

+

    ,'

    R+

    ,'

    ,nI/)min[Q()I,

    ,nl

    Nowchoosingtheargumentof/3in(3.3)andletting_1,wegetfor_zl:1andl,

    Equivalently z)+(

    Q(Rz)+

    ?

    ()i

    )"lRn-k~()" ()

    Qc+()?+()-+( whichimpliesforeveryrealorcomplexnumberwithIJ1,R>landJ.}=1,

    )

    2

    ()()

    +

    ()Q(z)l?+()-+()" ThisinconjunctionwithLemma3givesforlI1,R>1and:1,

    2+

    (

    +

    andthetheoremfollows

    (RP(z+1

    2

    ,}1)

    J

/,?一//,,./

    \,?-,,/,?一/

    +

    +

    ,Jn

    .l,,,,/2?一2

    ?

    1

    H

    +

    Rl

    lllllIlIJ__lIllI??_______l

    ,??,,??\一一

    <一?

    ?川?,J

    6K.K.DewanetalSomePolynomialInequalitiesjtheComplexDomam RefeFences

    1]Ankeny,N.C.andRivlin,J.,OnaTheoremofS.Bemstein,PacificJ.Math.,5(1955),849

    852.

    [2Aziz,A.,GrowthofPolynomialswhoseZerosareWithinorOutsideaCircle,Bul1.Austra1.

    Math.Soc.,

    35(1987),247256,

    3Aziz,A.andDawood,Q.M.,InequalitiesforaPolynomialanditsDerivative,J.Approx.Theo

    ry,54(1988),

    306313.

    4Jain,VK.,OnMaximumModulusofPolynomial,IndianJ.PureandAppliedMath.,23:l1(19

    92),815?819.

[5JJain,VK.,GeneralizationofCertainwellKnownInequalitiesforPolynomials,GlasnikM

    atematic'ki,

    32:52(1997),4551.

    6P61ya,G.andSzeg6,G.,AufgabenundLehrsatzeausderAnalysis,Springer-Verlag,Berlin,(

    1925).

    DepartmentOfMathematics

    FacultyofNaturalSciences

    JamiaMilliaIslamia

    CentralUniversity

    NewDelhi110025

    India

    K.K.Dewan

    E-mail:kkdewan123@yahoo.CO.in SunilHans

    Email:sunil.hans82@yahoo.com

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