Research Inventy : International Journal of Engineering and Science

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Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.

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Research Inventy : International Journal of Engineering and Science

  1. 1. Research Inventy: International Journal Of Engineering And ScienceIssn: 2278-4721, Vol. 2, Issue 6 (March 2013), Pp 06-10Www.Researchinventy.Com Zero-Free Regions for Polynomials With Restricted Coefficients M. H. Gulzar Department of Mathematics University of Kashmir, Srinagar 190006Abstract : According to a famous result of Enestrom and Kakeya, if P( z)  a n z n  a n1 z n1  ......  a1 z  a0is a polynomial of degree n such that 0  a n  a n1  ......  a1  a0 ,then P(z) does not vanish in z  1 . 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, Polynomials 1. Introduction And Statement Of Results The following elegant result on the distribution of zeros of a polynomial is due to Enestrom andKakeya [6] : nTheorem A : If P( z )  a j 0 j z j is a polynomial of degree n such that a n  a n1  ......  a1  a0  0 ,then P(z) has all its zeros in z  1 . n 1Applying the above result to the polynomial z P( ) , we get the following result : z nTheorem B : If P( z )  a j 0 j z j is a polynomial of degree n such that 0  a n  a n1  ......  a1  a0 ,then P(z) does not vanish in z  1 .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: nTheorem C: Let P( z )  a j 0 j z j be a polynomial of degree n . If for some real number k  1 , 0  a n  a n1  ......  a1  ka0 ,then P(z) does not vanish in the disk 1 . z  2k  1  ,0    a n , nTheorem D: Let P( z )   a j z j be a polynomial of degree n . If for some real number j 0 0  a n    a n1  ......  a1  a0 ,then P(z) does not vanish in the disk 6
  2. 2. Zero-free Regions for Polynomials with Restricted Coefficients 1 z  . 2 1 a0 nTheorem E: Let P( z )  aj 0 j z j be a polynomial of degree n . If for some real number k  1 , kan  a n1  ......  a1  a0  0 ,then P(z) does not vanish in a0 z . 2kan  a 0 nTheorem F: Let P( z )  a j 0 j z j be a polynomial of degree n . If for some real number   0, , an    an1  ......  a1  a0  0 ,then P(z) does not vanish in the disk a0 z  . 2(a n   )  a 0In this paper we give generalizations of the above mentioned results. In fact, we prove the following results: nTheorem 1: Let P( z )  a j 0 j z j be a polynomial of degree n . If for some real numbers k  1 and  0, , a n    a n1  ......  a1  ka0 ,then P(z) does not vanish in the disk a0 z . k (a 0  a 0 )  a 0  2   a n  a nRemark 1: Taking 0    a n , Theorem 1 reduces to Theorem C and taking k=1 and 0    an , , itreduces to Theorem D. nTheorem 2: Let P( z )  a j 0 j z j be a polynomial of degree n . If for some real numbers   0 and0    1, a n    a n1  ......  a1  a0 ,then P(z) does not vanish in a0 z . 2   a n  a n   (a 0  a 0 )  a 0Remark 2: Taking  1 and a 0  0 , Theorem 1 reduces to Theorem F and taking   1 , a0  0 and   (k  1)a n , k  1 , it reduces to Theorem E .Also taking   (k  1)a n , k  1 , we get the following result which reduces to Theorem E by taking a 0  0and   1 . nTheorem 3: Let P( z )  a j 0 j z j be a polynomial of degree n . If for some real numbers k  1,0    1 , kan  a n1  ......  a1  a0 ,then P(z) does not vanish in the disk 7
  3. 3. Zero-free Regions for Polynomials with Restricted Coefficients a0 z  . 2kan  (1  2 )a0 2.Proofs of the Theorems Proof of Theorem 1: We have P( z)  a n z n  a n1 z n1  ......  a1 z  a0 .Let 1 Q( z )  z n P ( ) zand F ( z)  ( z  1)Q( z) .Then F ( z )  ( z  1)(a0 z n  a1 z n1  ......  an1 z  an )  a0 z n1  [(a0  a1 ) z n  (a1  a 2 ) z n1  ......  (a n2  a n1 ) z 2  (a n1  a n ) z  a n ] .For z  1 , n 1 n 1 F ( z)  a0 z  [ a0  a1 z  a1  a 2 z  ......  a n1  a n z  a n ] n 1 a1  a 2 a n 1  a n an  a0 z [ z  { a 0  a1   ......   n }] n n 1 a0 z z z 1  a0 z [ z  { ka0  a1  ka0  a0  a1  a 2  ......  a n1  a n     n a0  an }] 1  a0 z [ z  {(ka0  a1 )  (k  1) a0  (a1  a2 )  ......  (a n 2  a n 1 ) n a0  (a n1  a n   )    a n }] 1  a0 z [ z  {k (a 0  a 0 )  a 0  a n  a n  2 }] n a0 0if 1 z  [k (a 0  a 0 )  a 0  a n  a n  2  ] . a0This shows that all the zeros of F(z) whose modulus is greater than 1 lie in the closed disk 1 z  [k (a 0  a 0 )  a 0  a n  a n  2  ] . a0But 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 1 z [k (a 0  a 0 )  a 0  a n  a n  2  ] . a0 1 1Since P( z )  z Q( ) , it follows, by replacing z by , that all the zeros of P(z) lie in n z z 8
  4. 4. Zero-free Regions for Polynomials with Restricted Coefficients a0 z . k (a 0  a 0 )  a 0  a n  a n  2 Hence P(z) does not vanish in the disk a0 z . k (a 0  a 0 )  a 0  a n  a n  2 That proves Theorem 1.Proof of Theorem 2: We have P( z)  a n z n  a n1 z n1  ......  a1 z  a0 .Let 1 Q( z )  z n P ( ) zand F ( z)  ( z  1)Q( z) .Then F ( z )  ( z  1)(a0 z n  a1 z n1  ......  an1 z  an )  a0 z n1  [(a0  a1 ) z n  (a1  a 2 ) z n1  ......  (a n2  a n1 ) z 2  (a n1  a n ) z  a n ] .For z  1 , n 1 n 1 F ( z)  a0 z  [ a0  a1 z  a1  a 2 z  ......  a n1  a n z  a n ] n 1 a1  a 2 a n 1  a n an  a0 z [ z  { a 0  a1   ......   n }] n n 1 a0 z z z 1  a0 z [ z  {a 0  a1  a 0  a 0  a1  a 2  ......  a n 1  a n     n a0  an }] 1  a0 z [ z  {(a1  a 0 )  (1   ) a 0  (a 2  a1 )  ......  (a n    a n 1 ) n a0    a n }] 1  a0 z [ z  { a 0   (a 0  a 0 )  a n  a n  2 }] n a0 0if 1 z  { a 0   (a 0  a 0 )  a n  a n  2 } . a0This shows that all the zeros of F(z) whose modulus is greater than 1 lie in the closed disk 1 z { a 0   (a 0  a 0 )  a n  a n  2 } . a0But 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 1 z { a 0   (a 0  a 0 )  a n  a n  2 } . a0 9
  5. 5. Zero-free Regions for Polynomials with Restricted Coefficients 1 1Since P( z )  z Q( ) , it follows, by replacing z by n , that all the zeros of P(z) lie in z z a0 z . a 0   (a 0  a 0 )  a n  a n  2 Hence P(z) does not vanish in the disk a0 z . a 0   (a 0  a 0 )  a n  a n  2 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, World Scientific , 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 and Engineering Applications, Vol. 6 No. IV(July 2012), 33-42. 10

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