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International Journal of Modern Engineering Research (IJMER)               www.ijmer.com         Vol.2, Issue.6, Nov-Dec. ...
International Journal of Modern Engineering Research (IJMER)               www.ijmer.com         Vol.2, Issue.6, Nov-Dec. ...
International Journal of Modern Engineering Research (IJMER)                www.ijmer.com         Vol.2, Issue.6, Nov-Dec....
International Journal of Modern Engineering Research (IJMER)                 www.ijmer.com         Vol.2, Issue.6, Nov-Dec...
International Journal of Modern Engineering Research (IJMER)               www.ijmer.com         Vol.2, Issue.6, Nov-Dec. ...
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On The Zeros of Polynomials

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On The Zeros of Polynomials

  1. 1. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4318-4322 ISSN: 2249-6645 On The Zeros of Polynomials M. H. Gulzar Department of Mathematics University of Kashmir, Srinagar, 190006, IndiaAbstract: In this paper we extend Enestrom -Kakeya theorem to a large class of polynomials with complexcoefficients by putting less restrictions on the coefficients . Our results generalise and extend many known results in thisdirection.(AMS) Mathematics Subject Classification (2010) : 30C10, 30C15Key-Words and Phrases: Polynomials, Zeros, Bounds I. Introduction and Statement of Results Let P(z) be a polynomial of degree n. A classical result due to Enestrom and Kakeya [9] concerning thebounds for the moduli of the zeros of polynomials having positive coefficients is often stated as in the followingtheorem(see [9]) : nTheorem A (Enestrom-Kakeya) : Let P( z )  a j 0 j z j be a polynomial of degree n whose coefficients satisfy 0  a1  a2  ......  an .Then P(z) has all its zeros in the closed unit disk z  1 . In the literature there exist several generalisations of this result (see [1],[3],[4],[8],[9]). Recently Aziz and Zargar [2]relaxed the hypothesis in several ways and proved nTheorem B: Let P( z )  a j 0 j z j be a polynomial of degree n such that for some k  1 , kan  an1  ......  a1  a0 .Then all the zeros of P(z) lie in kan  a0  a0 z  k 1  . anFor polynomials ,whose coefficients are not necessarily real, Govil and Rehman [6] proved the following generalisation ofTheorem A: nTheorem C: If P( z )  a j 0 j z j is a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n,such that  n   n1  ......  1   0  0 ,where  n  0 , then P(z) has all its zeros in n 2 z  1 ( )(  j ) . n j 0 More recently, Govil and Mc-tume [5] proved the following generalisations of Theorems B and C: nTheorem D: Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n.If for some k  1, k n   n1  ......  1   0 ,then P(z) has all its zeros in n k n   0   0  2  j j 0 z  k 1  . n www.ijmer.com 4318 | Page
  2. 2. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4318-4322 ISSN: 2249-6645 nTheorem E: Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n.If for some k  1, k n   n1  ......  1   0 ,then P(z) has all its zeros in n k n   0   0  2  j j 0 z  k 1  . nM.H.Gulzar [7] proved the following generalisations of Theorems D and E. nTheorem F: Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n.If for some real number   0 ,    n   n1  ......  1   0 ,then P(z) has all its zeros in the disk n    n   0   0  2  j  j 0 z  . n n nTheorem G: Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n.If for some real number   0 ,    n   n1  ......  1   0 ,then P(z) has all its zeros in the disk n    n   0   0  2  j  j 0 z  . n nThe aim of this paper is to give generalizations of Theorem F and G under less restrictive conditions on the coefficients.More precisely we prove the following : nTheorem 1: Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n. Iffor some real numbers   0 , 0    1,    n   n1  ......  1   0 ,then P(z) has all its zeros in the disk n    n   (  0   0 )  2  0  2  j  j 0 . z  n nRemark 1: Taking   1 in Theorem 1, we get Theorem F. Taking   (k  1) n ,   1,Theorem 1 reduces toTheorem D and taking   0 ,  0  0 and   1, we get Theorem C. Applying Theorem 1 to P(tz), we obtain the following result: nCorollary 1 : Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n.If for some real numbers   0 , 0    1 and t>0,   t  n  t n1 n1  ......  t1   0 , nthen P(z) has all its zeros in the disk www.ijmer.com 4319 | Page
  3. 3. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4318-4322 ISSN: 2249-6645 n   t n  n   (  0   0 )  2  0  2  j t j  j 0 z  . t n n 1 t n 1  n In Theorem 1 , if we take  0  0 , we get the following result: nCorollary 2 : Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n.If for some real numbers  0 ,0    1,    n   n1  ......  1   0  0 ,then P(z) has all its zeros in the disk n   2(1   ) 0  2  j  j 0 z  1 . n n If we take    n1   n  0 in Theorem 1, we get the following result: nCorollary 3 : Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n,such that  n   n1  ......  1   0  0 .Then P(z) has all its zeros in n  n 1  2(1   ) 0  2  j  n 1 j 0 z 1  . n n Taking   1 in Cor.3, we get the following result: nCorollary 4 : Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n,such that  n   n1  ......  1   0  0 .Then P(z) has all its zeros in n  n 1  2  j  n 1 j 0 z 1  . n n If we apply Theorem 1 to the polynomial –iP(z) , we easily get the following result: nTheorem 2: Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j , j=0,1,2,……,n.,If for some real numbers   0 , 0    1,    n   n1  ......  1   0 ,then P(z) has all its zeros in the disk n    n   (  0   0 )  2  0  2  j  j 0 z  . n n On applying Theorem 2 to the polynomial P(tz), one gets the following result: nCorollary 5 : Let P( z )  a j 0 j z j be a polynomial of degree n with Re( a j )   j and Im(a j )   j ,j=0,1,2,……,n.If for some real numbers   0 , 0    1 and t>0, www.ijmer.com 4320 | Page
  4. 4. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4318-4322 ISSN: 2249-6645   t n  n  t n1  n1  ......  t1   0 ,then P(z) has all its zeros in the disk n   t n  n   (  0   0 )  2  0  2  j t j  j 0 z  . t n n 1 t n 1 n II. Proofs of the TheoremsProof of Theorem 1.Consider the polynomialF ( z)  (1  z) P( z)  (1  z)(an z n  an1 z n1  .......  a1 z  a0 )  a n z n1  (a n  a n1 ) z n  ......  (a1  a0 ) z  a0  a n z n1  ( n   n1 ) z n  ......  ( 1   0 ) z   0  i n z n1  i(  n   n1 ) z n  ......  i(1   0 ) z  i 0   n z n1  z n  (    n   n1 ) z n  ( n1   n2 ) z n1  ......  (1   0 ) z   ( 0   0 ) z   0  i   n z n1  ( n   n1 ) z n  ......  (1   0 ) z   0 . Then   n z n 1  z n  (    n   n 1 ) z n  ( n 1   n 2 ) z n 1  ......  ( 1   0 ) z F (z )    ( 0 -  0 )z   0  i   n z n 1  (  n   n 1 ) z n  ......  ( 1   0 ) z   0   1 1    n z       n   n 1   0 n   1   0 n 1  n z z   z  n 1   (1   )   1   0   j   j 1 z n j   j 2     n z n1  ......  (1   0 ) z   0 .Thus , for z  1 , n   n z    (    n   n 1 )   0  ( n 1   n  2 )  ......  ( 2   1 )    F ( z)  z    ( 1   0 )  (1   )  0    n  (  n   0 )   (  j   j 1 ) j 1 n   n  z   n z       n   (  0   0 )  2  0  2  j     j 0   0if n  n z       n   (  0   0 )  2  0  2  j . j 1Hence all the zeros of F(z) whose modulus is greater than 1 lie inthe disk n    n   (  0   0 )  2  0  2  j  j 0 z  . n nBut those zeros of F(z) whose modulus is less than or equal to 1 already satisfy the above inequality. Hence it follows thatall the zeros of F(z) lie in the disk www.ijmer.com 4321 | Page
  5. 5. International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4318-4322 ISSN: 2249-6645 n    n   (  0   0 )  2  0  2  j  j 0 z  . n nSince all 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 n    n   (  0   0 )  2  0  2  j  j 0 z  . n nThis completes the proof of Theorem 1. REFERENCES[1] N. Anderson , E. B. Saff , R. S.Verga , An extension of the Enestrom- Kakeya Theorem and its sharpness, SIAM. Math. Anal. , 12(1981), 10-22.[2] A.Aziz and B.A.Zargar, Some extensions of the Enestrom-Kakeya Theorem , Glasnik Mathematiki, 31(1996) , 239-244.[3] K.K.Dewan, N.K.Govil, On the Enestrom-Kakeya Theorem, J.Approx. Theory, 42(1984) , 239-244.[4] R.B.Gardner, N.K. Govil, Some Generalisations of the Enestrom- Kakeya Theorem , Acta Math. Hungar Vol.74(1997), 125-134.[5] N.K.Govil and G.N.Mc-tume, Some extensions of the Enestrom- Kakeya Theorem , Int.J.Appl.Math. Vol.11,No.3,2002, 246-253.[6] N.K.Govil and Q.I.Rehman, On the Enestrom-Kakeya Theorem, Tohoku Math. J.,20(1968) , 126-136.[7] M.H.Gulzar, On the Location of Zeros of a Polynomial, Anal. Theory. Appl., vol 28, No.3(2012)[8] A. Joyal, G. Labelle, Q.I. Rahman, On the location of zeros of polynomials, Canadian Math. Bull.,10(1967) , 55-63.[9] M. Marden , Geometry of Polynomials, IInd Ed.Math. Surveys, No. 3, Amer. Math. Soc. Providence,R. I,1996.[10] G.V. Milovanoic , D.S. Mitrinovic and Th. M. Rassias, Topics in Polynomials, Extremal Problems, Inequalities, Zeros, World Scientific Publishing Co. Singapore,New York, London, Hong-Kong, 1994. www.ijmer.com 4322 | Page

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