M. H. Gulzar and A. W. Manzoor/ BIBECHANA 13 (2016) 1-8 : RCOST p.1 (Online Publication: Dec., 2015) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (Print), 2382-5340 (0nline) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Research Council of Science and Technology, Biratnagar, Nepal Generalizations of some Enestrom-Kakeya type results M.H. Gulzar*, A.W. Manzoor Department of Mathematics, University of Kashmir, Srinager-190006, Jammu and Kashmir, India. *E-mail: gulzarmh@gmail.com Article history: Received 05 June, 2015; Accepted 20 August, 2015 DOI: http://dx.doi.org/10.3126/bibechana.v13i0.13309 Abstract In this paper we give interesting generalizations of some well-known Enestrom-Kakeya type results on the location of zeros of a complex polynomial under less restrictive conditions on the coefficients of the polynomial. ©RCOST: All rights reserved. Keywords: Bound; Coefficient; Polynomial; Zeros. 1. Introduction Regarding the zeros of a polynomial with real and positive coefficients, we have the following result known as the Enestrom-Kakeya Theorem [1, 2, 3]. 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 )(zP lie in 1z . Various extensions and generalizations of this result are available in the literature. Joyal et al [4] proved 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 )(zP has all its zeros in the disk n n a aaa z 00   . As a generalization of Theorems A and B, Aziz and Zargar [5] proved the following: Theorem C: Let    n j j j zazP 0 )( be a polynomial of degree n such that for some 1k 011 ...... aaaka nn   . M. H. Gulzar and A. W. Manzoor/ BIBECHANA 13 (2016) 1-8 : RCOST p.2 (Online Publication: Dec., 2015) Then )(zP has all its zeros in the disk n n a aaka kz 001   . Shah et al [6] extended Theorem B to polynomials with complex coefficients and proved the following result: Theorem D: Let    n j j j zazP 0 )( be a complex polynomial of degree n with jja )Re( and jja )Im( for ,,......,2,1,0 nj  such that for some 1k .0...... ...... 011 011       nn nnk . Then )(zP has all its zeros in the disk n nn n n a k k a z    00)1( . Liman et al [7] proved the following generalization of Theorem D: Theorem E: Let    n j j j zazP 0 )( be a complex polynomial of degree n with jja )Re( and jja )Im( for 0,,......,2,1,0  nanj . If for some positive integer n and 1k ,............ 0111 2 2 1 1 1         kkkkk n n n n n n 0....... 011    nn , then all the zeros of )(zP lie in n n n j njjn n n a k k a z              )()1( )1( 00 . In the same paper they proved the following result also. Theorem F: Let    n j j j zazP 0 )( be a complex polynomial of degree n with jja )Re( and jja )Im( for 0,,......,2,1,0  nanj . If for some positive integer n , 1k and 1 , ,............ 0111 2 2 1 1 1         kkkkk n n n n n n 0....... 011    nn , then all the zeros of )(zP lie in n n n j njjn n nn a k a ik z                )()1( 1 00 . Recently Gulshan Singh [8] proved the following generalization of Theorems E and F: M. H. Gulzar and A. W. Manzoor/ BIBECHANA 13 (2016) 1-8 : RCOST p.3 (Online Publication: Dec., 2015) Theorem G: Let    n j j j zazP 0 )( be a complex polynomial of degree n with njia jjj ,......,2,1,0,   , where j and j are real numbers. If for some positive integers n, and for some real numbers 1,10,10 21  k , ,............ ,............ 02111 2 2 1 1 1 01111 2 2 1 1 1                 kkkkk kkkkk n n n n n n n n n n n n then all the zeros of )(zP lie in  )()1()1()()( 1 1 0002010201   nn na kz nn n j jj n j jjk      )()()1( . In this paper we prove the following result which not only generalizes the above results but also gives many other results for different values of the parameters. 2. Theorems and Proofs Theorem 1. Let    n j j j zazP 0 )( be a complex polynomial of degree n with njia jjj ,......,2,1,0,   , where j and j are real numbers. If for some positive integers n, and for some real numbers 1,1,10,10 2121  kk , ,............ ,............ 021121 2 22 1 212 1 2 011111 2 12 1 111 1 1                 kkkbkk kkkkk n n n n n n n n n n n n then all the zeros of )(zP lie in  010201 21 )1()()( 1)1()1(      nn nn nn aa kik z )()1( 0002   ||)1()()1( 1 n n j jj kk            n n j jj kk   )1(()1( 22          . Taking ,, 2121   kkk in Theorem 1 we get Theorem G. Taking 0,1,1, 02121   andkkk in Theorem 1, we get Theorem E. Taking 01,,, 02121   andnkkk in Theorem 1 we get Theorem F of Liman et al.Taking   n in Theorem 1,we get a Theorem of Gulzar [9,Theorem 1]. Taking 2121 1  kk in Theorem 1 we get the following interesting result: M. H. Gulzar and A. W. Manzoor/ BIBECHANA 13 (2016) 1-8 : RCOST p.4 (Online Publication: Dec., 2015) Corollary 1. Let    n j j j zazP 0 )( be a complex polynomial of degree n with ,,......,2,1,0)Im(.)Re( njforaa jjjj   such that ....... ,...... 011 011       nn nn . Then )(zP has all its zeros in the disk  nn na z   1 Taking ja real, that is j =0 for j=0,1,2,……..,n and 00  ,Corollary 1 reduces to Enestrom-Kakeya Theorem. Taking   in Theorem 1, we get the following result:. Corollary 2. Let    n j j j zazP 0 )( be a complex polynomial of degree n with 0,,......,2,1,0)Im(.)Re(  njjjj anjforaa  . If for some positive integer ,1,1,10,10, 2121  kkn  ,............ ,............ 021121 2 22 1 212 1 2 011111 2 12 1 111 1 1                 kkkkk kkkkk n n n n n n n n n n n n then all the zeros of )(zP lie in  010201 21 )1()()( 1)1()1(      nn nn nn aa kik z )()1( 0002   ||)1()()1( 1 n n j jj kk            n n j jj kk   )1(()1( 22          . Many other results may be deduced from Theorem 1 for different values of parameters. Proof of Theorem 1: Consider the polynomial M. H. Gulzar and A. W. Manzoor/ BIBECHANA 13 (2016) 1-8 : RCOST p.5 (Online Publication: Dec., 2015)         }]........)1( )1()()(........)( )()()( .......)()()1({ ........)1()1()()( .......)()()()( .......)()()1( ........... )(..........)( )(..........)( )...........)(1( )()1()( 1 1 1 12 002021 2 12 2 32 1 2112 1 12 1 212122 1 1 1 11001011 2 12 2 32 1 2111 1 11 1 211111 1 0011 0011 1 0011 1 01 1 1                                  zzzk zzzz zzkzk zkzkzki zzzkzzz zzzkzk zkzkzkza zzi zzza azaazaaza azazazaz zpzzF n n n nn n nn n n n n n nn n nn n n n n n nn n nn n n n nn n n n n n n                                             }] 1 ........ 1 )1( 1 )1( 1 )( 1 )(........ 1 )( 1 )( 1 )( 1 )( ....... 1 )(){( 1 ........ 1 )1( 11 )1( 1 )( 1 )(....... 1 )( 1 )( 1 )( 1 )( .......... 1 )()()1()1([ 12 0 102 1021212232 12112112 21212 110101 1011212232 12111111 2111121                                             nnnn nnn nnn nnnn nnnn nnn nnn nnnnnnn n zz k zz zzz zz k z k z kki zz k zz zzz zz k z k z kkkikzaz For |z|>1 so that nj z jn ,,.........2,1,0,1 || 1  .We have by using the hypothesis, M. H. Gulzar and A. W. Manzoor/ BIBECHANA 13 (2016) 1-8 : RCOST p.6 (Online Publication: Dec., 2015)  }] || 1 ||........ || 1 ||)1( || || || 1 |||1| || 1 || || 1 ||........ || 1 || || 1 || || 1 || || 1 || ....... || 1 |||| || 1 ||........ || 1 ||)1( || 1 || || 1 ||1 || 1 || || 1 || ....... || 1 || || 1 || || 1 || || 1 )1()1(||)( 12 0 102 1021212232 12112112 2121211 101011011212 23212111 2111121                                         nnnn nnn nnn nnnnnn nnnn nnn nnnnnnn n zz k zz zzz zz k z k z kk zz k zzzz zzz k z kkkikzazzF     ||........||)1(||||)1( ........ .......||........||)1( ||||)1( ....... )1()1(|| 12002 0211232 211212 2121211 001011 322111 2111121                          n nnnnn nnnnnnn n k kk kkk k kkkikzaz      0 )1(()1()1()()1(|||| ||)1(||)1()1()1(|| 221100 0201020121                  n n j jjn n j jj nnnnn n kkkk kikzaz    if   n n j jj n n j jj nnnnn kk kk kikza      )1(()1( )1(()1(|||| ||)1(||)1()()1()1( 22 1100 0201020121                    that is, if M. H. Gulzar and A. W. Manzoor/ BIBECHANA 13 (2016) 1-8 : RCOST p.7 (Online Publication: Dec., 2015)  )()1()1()()( 1)1()1( 0002010201 21      nn nn nn aa kik z n n j jj n n j jj kk kk     )1(()1( ||)1()()1( 22 1                   This shows that those zeros of F(z) whose modulus is greater than 1 lie in  )()1()1()()( 1)1()1( 0002010201 21      nn nn nn aa kik z n n j jj n n j jj kk kk     )1(()1( ||)1()()1( 22 1                   Since those zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality, it follows that all the zeros of F(z) and hence of p(z) lie in  )()1()1()()( 1)1()1( 0002010201 21      nn nn nn aa kik z n n j jj n n j jj kk kk     )1(()1( ||)1()()1( 22 1                   This completes the proof of Theorem 1. References [1] M. Marden, Geometry of Polynomials, Math. Surveys No.3, Amer. Math. Soc. Providence R.I. 1996. [2] G.V. Milovanovic, D.S. Mitrinovic and T.M. Rassias, Topics in Polynomials, Extremal Problems, Inequalities, Zeros. World Scientific Publishing Co. River Edge, NJ, 1994. [3] Q. I. Rahman and G. Schmeisser, Analytic Theory of polynomials, Oxford University Press, 2002. [4] A. Joyal, G. Labelle and Q.I. Rahman, Canad. Math. Bull. 10 (1967) 53-63. [5] A. Aziz and B. A. Zargar, Mathematiki, 31(1996) 239-244. M. H. Gulzar and A. W. Manzoor/ BIBECHANA 13 (2016) 1-8 : RCOST p.8 (Online Publication: Dec., 2015) [6] W. M. Shah and A. Liman, On Enestrom-kakeya Theorem and related Analytic Functions, Proc. Indian Acad. Sci (Math Sci) 17 ( 3) (2007)359-370. [7] A. Liman, Tawheeda Rasool and W.M. Shah, Bibechana, 10(2014) 71-81. [8] Gulshan Singh, American Journal of Mathematical Analysis, 2 (1) (2014) 15- 18. [9] M. H. 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