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# International Journal of Computational Engineering Research(IJCER)

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International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.

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### International Journal of Computational Engineering Research(IJCER)

1. 1. International Journal of Computational Engineering Research||Vol, 03||Issue, 11|| Polynomials having no Zero in a Given Region 1 M. H. Gulzar 1 Department of Mathematics University of Kashmir, Srinagar 190006 ABSTRACT: In this paper we consider some polynomials having no zeros in a given region. Our results when combined with some known results give ring –shaped regions containing a specific number of zeros of the polynomial. Mathematics Subject Classification: 30C10, 30C15 Keywords and phrases: Coefficient, Polynomial, Zero. I. INTRODUCTION AND STATEMENT OF RESULTS In the literature we find a large number of published research papers concerning the number of zeros of a polynomial in a given circle. For the class of polynomials with real coefficients, Q. G. Mohammad [5] proved the following result:  Theorem A: Let P( z )   a j z j be a polynomial o f degree n such that j 0 an  an1  ......  a1  a0  0 . 1 Then the number of zeros of P(z) in z  does not exceed 2 a 1 log n . log 2 a0 Bidkham an d Dewan [1] generalized Theorem A in the following way: 1  Theorem B: Let P( z )   a j z j be a polynomial o f degree n such that j 0 a n  a n1  ......  a k 1  a k  a k 1  ......  a1  a0  0 , 1 for some k ,0  k  n .Then the number of zeros of P(z) in z  does not exceed 2  a  a0  a n  a0  2 a k 1  log  n log 2 a0   .     Ebadian et al [2] generalized the above results by proving the following results:  Theorem C: Let P( z )   a j z j be a polynomial o f degree n such that j 0 a n  a n1  ......  a k 1  a k  a k 1  ......  a0 for some k, 0  k  n .Then the number of zeros of P(z) in z  R ,R>0, does not exceed 2  a n R n 1  a 0  R k (a k  a 0 )  R n (a k  a n )  1   for R  1 log   log 2 a0     || Issn 2250-3005 || || November || 2013 || Page 8
2. 2. Polynomials having no Zero in a Given Region And  a n R n 1  a 0  R( a k  a 0 )  R n ( a k  a n )  1   log   log 2 a0     for R 1 . M.H.Gulzar [3] generalized the above result by proving the following result:  Theorem D: Let P( z )   a j z j be a polynomial o f degree n with Re( a j )   j , Im(a j )   j such that j 0 k , ,  ,0  k  1,0    1, 0    n , k n   n1  ......    1       1  ......   0 . R Then the number of zeros of P(z) in z  ( R  0, c  1) does not exceed c for some  1  n 1  a 0  R  [    (  0   0 )   0   0     2  j  an R j 1  n 1  n  R [ n  k( n   n )       n  2   j ] 1  j   1 log  log c a0       for R 1  ]            and  1  a n R n 1  a 0  R[    (  0   0 )   0   0     2  j  j 1  n 1  n  R [ n  k( n   n )       n  2   j ] 1  j   1 log  log c a0        ]            In this paper we prove the following result:  Theorem 1: Let P( z )   a j z j be a polynomial o f degree n with Re( a j )   j , Im(a j )   j such that j 0 for some k , ,  ,0  k  1,0    1, 0    n , k n   n1  ......    1       1  ......   0 . Then P(z) has no zero in z a0 M1 for R  1 and no zero in z  a0 for M2 R 1 where n M 1  an R n1  R n [  n  k (  n   n )    ]      n  2   j ] j   1  1  R  [    (  0   0 )   0   0     2  j ] j 1 and n M 2  an R n1  R n [  n  k (  n   n )    ]      n  2   j ] j   1  1  R[    (  0   0 )   0   0     2  j ] . j 1 || Issn 2250-3005 || || November || 2013 || Page 9
3. 3. Polynomials having no Zero in a Given Region Combining Theorem 1 with Theorem D, we get the following result:  Theorem 2: Let P( z )   a j z j be a polynomial o f degree n with Re( a j )   j , Im(a j )   j such that j 0 for some k , ,  ,0  k  1,0    1, 0    n , k n   n1  ......    1       1  ......   0 . a0 Then the number of zeros of P(z) in M1  z R ( R  0, c  1) does not exceed c  1  a n R n 1  a 0  R  [    (  0   0 )   0   0     2  j  j 1  n 1  n  R [ n  k( n   n )       n  2   j ] 1  j   1 log  log c a0       for and the number of zeros of P(z) in a0 M2  z R 1  ]            R ( R  0, c  1) does not exceed c  1  n 1  a 0  R[    (  0   0 )   0   0     2  j  an R j 1  n 1  n  R [ n  k( n   n )       n  2   j ] 1  j   1 log  log c a0       for R 1 ,  ]            where M 1 and M 2 are as given in Theorem 1. For different values of the parameters, we get many interesting results including some already known results. 2. Proofs of Theorems Proof of Theorem 1: Consider the polynomial F(z) =(1-z)P(z)  (1  z )(a n z n  a n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  a0  [(k n   n 1 )  (k  1) n ]z n  n 1  (  j  1  j   ( j   j 1 ) z j  [( 1   0 )  (  1) 0 ]z  i{ j 2   j 1 ) z j n  (  j  1 j   j 1 ) z j    (  j   j 1 ) z j } j 1  a 0  G( z ) , where || Issn 2250-3005 || || November || 2013 || Page 10
4. 4. Polynomials having no Zero in a Given Region G( z )  a n z n 1  a0  [(k n   n 1 )  (k  1) n ]z n  n 1  (  j  1   j 2   j 1 ) z j j j 1   ( j   j 1 ) z j  [( 1   0 )  (  1) 0 ]z  i  (  j   j 1 ) z j z  R , we have, by using the hypothesis For G( z )  a n R n 1  [( n 1  k n )  (1  k )  n ]R n  n 1  (  j  1 j 1   j )R j  n j 1 j 1   ( j   j 1 ) R j  [( 1   0 )  (1   )  0 ]R   (  j   j 1 ) R j which gives n G( z )  an R n1  R n [  n  k (  n   n )    ]      n  2   j ] j   1  1  R  [    (  0   0 )   0   0     2  j ] j 1  M1 for R 1 and n G( z )  an R n1  R n [  n  k (  n   n )    ]      n  2   j ] j   1  1  R[    (  0   0 )   0   0     2  j ] j 1  M2 R  1. Since G(z) is analytic in z  R and G(0)=0, it follows by Schwarz Lemma that for G( z )  M 1 z for R  1and G( z )  M 2 z for R  1. Hence, for R  1 , F ( z )  a0  G( z )  a0  G( z )  a0  M 1 z 0 if z  a0 . M1 R  1, F ( z )  a0  G( z ) And for  a0  G( z )  a0  M 2 z 0 if z  a0 . M2 This shows that F(z) has no zero in z a0 M1 for R  1 and no zero in z  a0 M2 for R  1 . But the zeros of P(z) are also the zeros of F(z). Therefore, the result follows. || Issn 2250-3005 || || November || 2013 || Page 11
5. 5. Polynomials having no Zero in a Given Region REFERENCES [1] [2] [3] [4] [5] M.Bidkham and K.K. Dewan, On the zeros of a polynomial, Numerical Methods and Approximation Theory,III , Nis (1987), 121128. A.Ebadian, M.Bidkham and M.Eshaghi Gordji, Number of zeros of a polynomial in a given domain,Tamkang Journal of Mathematics, Vol.42, No.4,531-536, Winter 2011. M.H.Gulzar, On the Number of Zeros of a Polynomial in a Given Domain, International Journal of Scientific and Research Publications, Vol.3, Issue 10, October 2013. M.Marden , Geometry of Polynomials , Mathematical surveys, Amer. Math. Soc. , Rhode Island , 3 (1966). Q.G.Mohammad, Location of the zeros of polynomials, Amer. Math. Monthly , 74 (1967), 290-292. || Issn 2250-3005 || || November || 2013 || Page 12