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Bounds and a majorization for the real parts of the zeros of polynomials


Author: Fuad Kittaneh
Journal: Proc. Amer. Math. Soc. 135 (2007), 659-664
MSC (2000): Primary 15A18, 15A42, 26C10, 30C15
DOI: https://doi.org/10.1090/S0002-9939-06-08509-1
Published electronically: August 31, 2006
MathSciNet review: 2262860
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Abstract: We apply some eigenvalue inequalities to the real parts of the Frobenius companion matrices of monic polynomials to establish new bounds and a majorization for the real parts of the zeros of these polynomials.


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  • [1] A. A. Abdurakhmanov, The geometry of the Hausdorff domain in localization problems for the spectrum of arbitrary matrices, Math. USSR-Sb. 59 (1988), 39-51. MR 0868600 (88e:47010)
  • [2] Y. A. Alpin, M.-T. Chien and L. Yeh, The numerical radius and bounds for zeros of a polynomial, Proc. Amer. Math. Soc. 131 (2003), 725-730. MR 1937409 (2003h:26021)
  • [3] R. Bhatia, Matrix Analysis, Springer, New York, 1997. MR 1477662 (98i:15003)
  • [4] M. Fujii and F. Kubo, Operator norms as bounds for roots of algebraic equations, Proc. Japan Acad. Sci. 49 (1973), 805-808. MR 0364310 (51:565)
  • [5] M. Fujii and F. Kubo, Buzano's inequality and bounds for roots of algebraic equations, Proc. Amer. Math. Soc. 117 (1993), 359-361. MR 1088441 (93d:47014)
  • [6] K. E. Gustafson and D. K. M. Rao, Numerical Range, Springer, New York, 1997. MR 1417493 (98b:47008)
  • [7] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985. MR 0832183 (87e:15001)
  • [8] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991. MR 1091716 (92e:15003)
  • [9] F. Kittaneh, Singular values of companion matrices and bounds on zeros of polynomials, SIAM J. Matrix Anal. Appl. 16 (1995), 333-340. MR 1311437 (95m:15015)
  • [10] F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158 (2003), 11-17. MR 2014548 (2004i:15022)
  • [11] F. Kittaneh, Bounds for the zeros of polynomials from matrix inequalities, Arch. Math. (Basel) 81 (2003), 601-608. MR 2029723 (2004j:15035)
  • [12] H. Linden, Bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices, Linear Algebra Appl. 271 (1998), 41-82. MR 1485162 (98m:65059)
  • [13] H. Linden, Numerical radii of some companion matrices and bounds for the zeros of polynomials, in: Analytic and Geometric Inequalities and Applications, Math. Appl. 478, Kluwer, Dordrecht, 1999, 205-229. MR 1785871 (2001i:15031)
  • [14] M. Marden, Geometry of Polynomials, 2nd ed., Math. Surveys 3, Amer. Math. Soc., Providence, RI, 1966. MR 0225972 (37:1562)
  • [15] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979. MR 0552278 (81b:00002)
  • [16] G. Schmeisser, Sharp inequalities for the zeros of polynomials and power series, Result. Math. 39 (2001), 333-344. MR 1834579 (2002c:30011)

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Additional Information

Fuad Kittaneh
Affiliation: Department of Mathematics, University of Jordan, Amman, Jordan
Email: fkitt@ju.edu.jo

DOI: https://doi.org/10.1090/S0002-9939-06-08509-1
Keywords: Frobenius companion matrix, zeros of polynomials, eigenvalue, majorization
Received by editor(s): August 9, 2004
Received by editor(s) in revised form: September 28, 2005
Published electronically: August 31, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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