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### B02606010

1. 1. Research Inventy: International Journal Of Engineering And ScienceIssn: 2278-4721, Vol. 2, Issue 6 (March 2013), Pp 06-10Www.Researchinventy.Com6Zero-Free Regions for Polynomials With Restricted CoefficientsM. H. GulzarDepartment of Mathematics University of Kashmir, Srinagar 190006Abstract : According to a famous result of Enestrom and Kakeya, if0111 ......)( azazazazP nnnn  is a polynomial of degree n such that011 ......0 aaaa nn   ,then P(z) does not vanish in 1z . In this paper we relax the hypothesis of this result in several ways andobtain zero-free regions for polynomials with restricted coefficients and thereby present some interestinggeneralizations and extensions of the Enestrom-Kakeya Theorem.Mathematics Subject Classification: 30C10,30C15Key-Words and Phrases: Zero-free regions, Bounds, Polynomials1. Introduction And Statement Of ResultsThe following elegant result on the distribution of zeros of a polynomial is due to Enestrom andKakeya [6] :Theorem A : If njjj zazP0)( is a polynomial of degree n such that0...... 011   aaaa nn ,then P(z) has all its zeros in 1z .Applying the above result to the polynomial )1(zPzn, we get the following result :Theorem B : If njjj zazP0)( is a polynomial of degree n such that011 ......0 aaaa nn   ,then P(z) does not vanish in 1z .In the literature [1-5, 7,8], there exist several extensions and generalizations of the Enestrom-Kakeya Theorem .Recently B. A. Zargar [9] proved the following results:Theorem C: Let njjj zazP0)( be a polynomial of degree n . If for some real number 1k ,011 ......0 kaaaa nn   ,then P(z) does not vanish in the disk121kz .Theorem D: Letnjjj zazP0)( be a polynomial of degree n . If for some real number na  0, ,011 ......0 aaaa nn   ,then P(z) does not vanish in the disk
2. 2. Zero-free Regions for Polynomials with Restricted Coefficients70211az .Theorem E: Let njjj zazP0)( be a polynomial of degree n . If for some real number 1k ,0...... 011   aaaka nn ,then P(z) does not vanish in002 akaazn  .Theorem F: Let njjj zazP0)( be a polynomial of degree n . If for some real number ,0 ,0...... 011   aaaa nn  ,then P(z) does not vanish in the disk00)(2 aaazn .In this paper we give generalizations of the above mentioned results. In fact, we prove the following results:Theorem 1: Let njjj zazP0)( be a polynomial of degree n . If for some real numbers 1k and,0 ,011 ...... kaaaa nn   ,then P(z) does not vanish in the disknn aaaaakaz2)( 0000.Remark 1: Taking ,0 na  Theorem 1 reduces to Theorem C and taking k=1 and ,0 na  , itreduces to Theorem D.Theorem 2: Let njjj zazP0)( be a polynomial of degree n . If for some real numbers 0 and10  ,011 ...... aaaa nn    ,then P(z) does not vanish in0000)(2 aaaaaaznn .Remark 2: Taking 1 and 00 a , Theorem 1 reduces to Theorem F and taking 1 , 00 a and1,)1(  kak n , it reduces to Theorem E .Also taking 1,)1(  kak n , we get the following result which reduces to Theorem E by taking 00 aand 1 .Theorem 3: Let njjj zazP0)( be a polynomial of degree n . If for some real numbers 10,1  k ,011 ...... aaaka nn   ,then P(z) does not vanish in the disk
3. 3. Zero-free Regions for Polynomials with Restricted Coefficients800)21(2 akaazn  .2.Proofs of the TheoremsProof of Theorem 1: We have0111 ......)( azazazazP nnnn   .Let)1()(zPzzQ nand)()1()( zQzzF  .Then)......)(1()( 1110 nnnnazazazazzF  2121211010 )(......)()[( zaazaazaaza nnnnn])( 1 nnn azaa   .For 1z ,]......[)( 11211010 nnnnnnazaazaazaazazF  }]......{1[ 11211000 nnnnnnzazaazaaaaazza     nnnaaaaakaakaazza 121001000 ......{1[}]na)(......)()1(){(1[ 122101000   nnnaaaaakakaazza}])( 1 nnn aaa   }]2)({1[ 00000  nnnaaaaakazza0if]2)([10000 nn aaaaakaz .This shows that all the zeros of F(z) whose modulus is greater than 1 lie in the closed disk]2)([10000 nn aaaaakaz .But those zeros of F(z) whose modulus is less than or equal to 1 already lie in the above disk. Therefore, itfollows that all the zeros of F(z) and hence Q(z) lie in]2)([10000 nn aaaaakaz .Since )1()(zQzzP n , it follows, by replacing z byz1, that all the zeros of P(z) lie in
4. 4. Zero-free Regions for Polynomials with Restricted Coefficients92)( 0000nn aaaaakaz .Hence P(z) does not vanish in the disk2)( 0000nn aaaaakaz .That proves Theorem 1.Proof of Theorem 2: We have0111 ......)( azazazazP nnnn   .Let)1()(zPzzQ nand)()1()( zQzzF  .Then)......)(1()( 1110 nnnnazazazazzF  2121211010 )(......)()[( zaazaazaaza nnnnn])( 1 nnn azaa   .For 1z ,]......[)( 11211010 nnnnnnazaazaazaazazF  }]......{1[ 11211000 nnnnnnzazaazaaaaazza     nnnaaaaaaaaazza 121001000 ......{1[}]na)(......)()1(){(1[ 11200100  nnnaaaaaaaazza }]na }]2)({1[ 00000   nnnaaaaaazza0if}2)({10000  nn aaaaaaz .This shows that all the zeros of F(z) whose modulus is greater than 1 lie in the closed disk}2)({10000  nn aaaaaaz .But those zeros of F(z) whose modulus is less than or equal to 1 already lie in the above disk. Therefore, itfollows that all the zeros of F(z) and hence Q(z) lie in}2)({10000  nn aaaaaaz .
5. 5. Zero-free Regions for Polynomials with Restricted Coefficients10Since )1()(zQzzP n , it follows, by replacing z byz1, that all the zeros of P(z) lie in 2)( 0000nn aaaaaaz .Hence P(z) does not vanish in the disk 2)( 0000nn aaaaaaz .That proves Theorem 2.References[1] Aziz, A. and Mohammad, Q. G., Zero-free regions for polynomials and some generalizations of Enesrtrom-Kakeya Theorem,Canad. Math. Bull., 27(1984), 265-272.[2] Aziz, A. and Zargar B.A., Some Extensions of Enesrtrom-Kakeya Theorem, Glasnik Mathematicki, 31(1996) , 239-244.[3] Dewan, K. K., and Bidkham, M., On the Enesrtrom-Kakeya Theorem I., J. Math. Anal. Appl., 180 (1993) , 29-36.[4] Kurl Dilcher, A generalization of Enesrtrom-Kakeya Theorem, J. Math. Anal. Appl., 116 (1986) , 473-488.[5] Joyal, A., Labelle, G..and Rahman, Q. I., On the Location of zeros of polynomials, Canad. Math. Bull., 10 (1967), 53-63.[6] Marden, M., Geometry of Polynomials, IInd. Edition, Surveys 3, Amer. Math. Soc., Providence, (1966) R.I.[7] Milovanoic, G. V., Mitrinovic , D.S., Rassiass Th. M., Topics in Polynomials, Extremal problems, Inequalities , Zeros, WorldScientific , Singapore, 1994.[8] Rahman, Q. I., and Schmeisser, G., Analytic Theory of Polynomials, 2002, Clarendon Press, Oxford.[9] Zargar, B. A., Zero-free regions for polynomials with restricted coefficients, International Journal of Mathematical Sciences andEngineering Applications, Vol. 6 No. IV(July 2012), 33-42.