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# International Journal of Engineering Research and Development

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### International Journal of Engineering Research and Development

1. 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. 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. 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. 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. 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. 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. 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.