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International Journal of Engineering Research and Development
International Journal of Engineering Research and Development
International Journal of Engineering Research and Development
International Journal of Engineering Research and Development
International Journal of Engineering Research and Development
International Journal of Engineering Research and Development
International Journal of Engineering Research and Development
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International Journal of Engineering Research and Development

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  • 1. International Journal of Engineering Research and Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com Volume 10, Issue 5 (May 2014), PP.75-81 75 On Some Extensions of Enestrom-Kakeya Theorem M. H. Gulzar Department of Mathematics University of Kashmir, Srinagar 19000 Abstract: In this paper we generalize some recent extensions of the Enestrom-Kakeya Theorem concerning the location of zeros of polynomials. Mathematics Subject Classification: 30 C 10, 30 C 15 Keywords and Phrases: Coefficient, Polynomial, Zero I. INTRODUCTION AND STATEMENT OF RESULTS The following result known as the Enestrom-Kakeya Theorem [7] is an elegant result in the theory of distribution of the zeros of a polynomial: Theorem A: Let   n j j j zazP 0 )( be a polynomial of degree n such that 0...... 011   aaaa nn . Then all the zeros of P(z) lie in the closed disk 1z . In the literature [1-8], there exist several extensions and generalizations of Theorem A. Joyal, Labelle and Rahman [5] gave the following generalization of Theorem A: Theorem B: Let   n j j j zazP 0 )( be a polynomial of degree n such that 011 ...... aaaa nn   . Then all the zeros of P(z) lie in the closed disk n n a aaa z 00   . Aziz and Zargar [1] generalized Theorems A and B by proving the following results: Theorem C: Let   n j j j zazP 0 )( be a polynomial of degree n such that for some ,1k 011 ...... aaaka nn   . Then all the zeros of P(z) lie in the closed disk n n a aaka kz 00 1   . Recently, Zargar [8] gave the following generalizations of the above mentioned results: Theorem D: Let   n j j j zazP 0 )( be a polynomial of degree n 2 such that for some ,1k either 0....... 132   aaaka nn and 0....... 0231   aaaa nn , if n is odd or 0....... 022   aaaka nn and 0....... 1331   aaaa nn , if n is even. Then all the zeros of P(z) lie in the region n n a a kzz 1   ,
  • 2. On Some Extensions of Enestrom-Kakeya Theorem 76 where , are the roots of the quadratic 0112   kz a a z n n . Theorem E: Let   n j j j zazP 0 )( be a polynomial of degree n 2 such that either 0.............. 1312122   aaaaaa nn  and 0............. 0222231   aaaaaa nn  , for some integer 2 1 0,   n  , if n is odd or 0............. 022222   aaaaaa nn  and 0....... 13121231   aaaaaa nn  , for some integer 2 2 0,   n  , if n is even. Then all the zeros of P(z) lie in the closed disk n n n n a aaaaa a a z )2(2 1 1221011     In this paper, we give generalizations of Theorems D and E and prove the following results: Theorem 1: Let   n j j j zazP 0 )( be a polynomial of degree n 2 such that for some 1,0;1, 2121  kk , either 0....... 11321   aaaak nn  and 0....... 022312   aaaak nn  , if n is odd or 0....... 01221   aaaak nn  and 0....... 123312   aaaak nn  , if n is even. Then for odd n all the zeros of P(z) lie in the region n nn a aaakak zz 0211121 )1(2)1(2)12(      and for even n all the zeros of P(z) lie in the region n nn a aaakak zz 1201121 )1(2)1(2)12(      , where , are the roots of the quadratic 011 12   kz a a z n n . Remark 1: For 1,1,1, 2121  kkk , Theorem 1 reduces to Theorem D of Zargar. Taking 12 11  kaa nn , 12 k and noting that the quadratic 0112 11 2  kzkz has two equal roots each equal to 11  k ,we get the following Corollary 1: Let   n j j j zazP 0 )( be a polynomial of degree n 2 such that for some 1,0;1 211  k , either 0....... 11321   aaaak nn  and
  • 3. On Some Extensions of Enestrom-Kakeya Theorem 77 0.......12 022311   aaaaka nnn  , if n is odd, or 0....... 01221   aaaak nn  and 0.......12 123311   aaaaka nnn  , if n is even. Then for odd n all the zeros of P(z) lie in the region 2 1 021111 1 )1(2)1(212 1           n nn a aakaak kz  and for even n all the zeros of P(z) lie in the region 2 1 120111 1 )1(2)1(212 1           n nn a aakaak kz  . For 1,1, 211  kk , Cor. 1 reduces to a result of Zargar [8, Cor.1] . Applying Cor. 1 to a polynomial of even degree, we get the following Corollary 2: Let   n j j j zbzQ 2 0 )( be a polynomial of even degree 2n such that for some 1,0;1 211  k , 0....... 0122221   bbbbk nn  and 0.......12 123321221   bbbbbk nnn  . Then all the zeros of Q(z) lie in 2 1 2 12011221 1 )1(2)1(212 1           n nn b bbkbbk kz  . Taking 1,1,1 211  k , Cor. 2 reduces to Enestrom-Kakeya Theorem (see [8]). Taking ,21 k we get the following result from Cor.1: Corollary 3: Let   n j j j zazP 0 )( be a polynomial of degree n 2 such that for some 1,0 21   , either 0.......2 1132   aaaa nn  and 0.......2 02231   aaaaa nnn  , if n is odd, or 0.......2 0122   aaaa nn  and 0.......2 12331   aaaaa nnn  , if n is even. Then for odd n all the zeros of P(z) lie in the region 2 1 0211 )1(2)1(22 1         na aa z  and for even n all the zeros of P(z) lie in the region 2 1 1201 )1(2)1(22 1         na aa z  .
  • 4. On Some Extensions of Enestrom-Kakeya Theorem 78 For 1,1 21   , Cor.3 reduces to a result of Zargar [8,Cor.3]. Theorem 2: Let   n j j j zazP 0 )( be a polynomial of degree n 2 such that for some 1,0;1, 2121  kk , either 0.............. 113121221   aaaaaak nn  and 0............. 022222312   aaaaaak nn  , for some integer 2 1 0,   n  , if n is odd or 0............. 01222221   aaaaaak nn  and 0....... 1231212312   aaaaaak nn  , for some integer 2 2 0,   n  , if n is even. Then for odd n all the zeros of P(z) lie in the disk n nn n n a aaaaakak a a z 02111221211 2222)12()12(     and for even n all the zeros of P(z) lie in the disk n nn n n a aaaaakak a a z 12011221211 2222)12()12(     . Remark 2: For 1,1,1,1 2121  kk , Theorem 2 reduces to Theorem E of Zargar. Applying Theorem 2 to the polynomial P(tz), we get the following result: Corollary 4: Let   n j j j zazP 0 )( be a polynomial of degree n 2 such that for some 1,0;1, 2121  kk and t>0, either 0.............. 11 3 3 12 12 12 12 2 21        tatatatatatak n n n n     and 0............. 02 2 2 22 22 2 2 3 3 1 12        atatatatatak n n n n     , for some integer 2 1 0,   n  , if n is odd or 0............. 01 2 2 22 22 2 2 2 21      atatatatatak n n n n     and 0............. 12 3 3 12 12 12 12 3 3 1 12          tatatatatatak n n n n     , for some integer 2 2 0,   n  , if n is even. Then for odd n all the zeros of P(z) lie in the disk n n n n n n n n at atatatataktak a a z 1 0211 2 12 2 2 1 12 1 11 2222)12()12(             and for even n all the zeros of P(z) lie in the disk n n n n n n n n at taatatataktak a a z 1 1201 2 12 2 2 1 12 1 11 2222)12()12(             . For 1,1,1,1 2121  kk , Cor. 4 reduces to a result of Zargar [8, Cor.4]. 2. Proofs of Theorems
  • 5. On Some Extensions of Enestrom-Kakeya Theorem 79 Proof of Theorem 1: Suppose n is odd. Consider the polynomial )......)(1()()1()( 01 1 1 22 azazazazzPzzF n n n n    1 312121 1 1 2 0022 1 1 2 )()1()( )(......)(            n nn n n n nn n n n n n nn n n n n zaakzakzaakzaza azaazaazaza 01 2 02 2 022 3 11 3 113 2 222 12 1212 22 222 1 12 )1( )()1()(......)( )()(......)1( azaza zaazazaazaa zaazaazak n n                  For 1z , so that nj z j ,......,2,1,1 1  , we have , by using the hypothesis, z ak z aak aakakzazazzF nnn nnnnn n 12312 2111 2 )1( {)1([)(       }] )1()1( ............ 0 1 1 2 02 2 202 3 11 3 311 2 222 12 1212 22 222 nnnnn nnnn z a z a z a z aa z a z aa z aa z aa z aa                               123122111 2 )1({)1([   nnnnnnn n akaakaakakzazaz }])1()1( ............ 010202211 11322221212222 aaaaaa aaaaaaaaa      1112111 2 )1(2)12({)1([ aakakakzazaz nnnnn n    if 021112111 2 )1(2)1(2)12()1( aaakakakzaza nnnnn    02 )1(2 a  or ])1(2)1(2)12([ 1 1 02111211 12 aaakak a kz a a z nn nn n     This shows that all those zeros of F(z) whose modulus is greater than 1 lie in ])1(2)1(2)12([ 1 1 02111211 12 aaakak a kz a a z nn nn n     or ])1(2)1(2)12([ 1 0211121 aaakak a zz nn n    where , are the roots of the quadratic 011 12   kz a a z n n . Since the zeros of F(z) with modulus less than or equal to 1 already satisfy the above inequality, it follows that all the zeros of F(z)lie in ])1(2)1(2)12([ 1 0211121 aaakak a zz nn n    .
  • 6. On Some Extensions of Enestrom-Kakeya Theorem 80 Since the zeros of P(z) are also the zeros of F(z), , it follows that all the zeros of P(z)lie in ])1(2)1(2)12([ 1 0211121 aaakak a zz nn n    . The case of even n follows similarly. Proof of Theorem 2: Suppose n is odd. Consider the polynomial )......)(1()()1()( 01 1 1 22 azazazazzPzzF n n n n    1 312121 1 1 2 0022 1 1 2 )()1()( )(......)(            n nn n n n nn n n n n n nn n n n n zaakzakzaakzaza azaazaazaza 01 2 02 2 022 3 11 3 113 2 222 12 1212 22 222 1 12 )1( )()1()(......)( )()(......)1( azaza zaazazaazaa zaazaazak n n                  For 1z , so that 1,......,2,1,1 1  nj z j , we have , by using the hypothesis, 2 12 2 312121 1 1 )1()1( {[)( z ak z aak z ak z aak azazzF nnnnnn nn n            }] )1()1( ............ 1 01 1 02 1 202 2 11 2 311 12 222 2 1212 12 222                       nnnnn nnnn z a z a z a z aa z a z aa z aa z aa z aa         123121211 1 )1()1({[    nnnnnnnn n akaakakaakazaz }])1()1( ............ 010220211 3112221212222 aaaaaa aaaaaaaa      1221211 1 22)12()12{([     aaakakazaz nnnn n }]22 0211 aa   0 if 02111221211 2222)12()12( aaaaakakaza nnnn    i.e. n nn n n a aaaaakak a a z 02111221211 2222)12()12(     This shows that all those zeros of F(z) whose modulus is greater than 1 lie in n nn n n a aaaaakak a a z 02111221211 2222)12()12(     . Since the zeros of F(z) with modulus less than or equal to 1 already satisfy the above inequality, it follows that all the zeros of F(z) lie in the disk n nn n n a aaaaakak a a z 02111221211 2222)12()12(     . Since the zeros of P(z) are also the zeros of F(z), , it follows that all the zeros of P(z)lie in the disk
  • 7. On Some Extensions of Enestrom-Kakeya Theorem 81 n nn n n a aaaaakak a a z 02111221211 2222)12()12(     . The case of even n follows similarly. REFERENCES [1] A.Aziz and B.A.Zargar, Some Extensions of Enestrom-Kakeya Theorem, Glasnik Math. 31(1996), 239-244. [2] N. K. Govil and Q.I. Rahman, On the Enestrom-Kakeya Theorem, Tohoku Math.J.20(1968), 126-136. [3] M. H. Gulzar, Some Refinements of Enestrom-Kakeya Theorem, Int. Journal of Mathematical Archive -2(9, 2011),1512-1529. [4] M. H. Gulzar, Bounds for the Zeros and Extremal Properties of Polynomials, Ph.D thesis, Department of Mathematics, University of Kashmir, Srinagar, 2012. [5] A. Joyal, G. Labelle and Q. I. Rahman, On the Location of Zeros of Polynomials, Canad. Math. Bull., 10(1967), 53-66. [6] A. Liman and W. M. Shah, Extensions of Enestrom-Kakeya Theorem, Int. Journal of Modern Mathematical Sciences, 2013, 8(2), 82-89. [7] M. Marden, Geometry of Polynomials, Math. Surveys, No.3; Amer. Math. Soc. Providence R.I. 1966. [8] B.A.Zargar, On Enestrom-Kakeya Theorem, Int. Journal of Engineering Science and Research Technology, 3(4), April 2014.

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