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Some Bounds for the Zeros of Polynomials

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In this paper, we find a bound,for all the zeros of a polynomial interms of its coefficents similar to the bound given by Montel (1932) and Kuneyida(1916) as an improvement of Cauchy's classical theorem. In fact, we use a generalized version of Holders inequality for obtaining various interesting bounds for all the zeros of a polynomial as a function of their coefficients.

Abstract

In this paper, we find a bound,for all the zeros of a polynomial interms of its coefficents similar to the bound given by Montel (1932) and Kuneyida(1916) as an improvement of Cauchy's classical theorem. In fact, we use a generalized version of Holders inequality for obtaining various interesting bounds for all the zeros of a polynomial as a function of their coefficients.

참고문헌 (14)

  1. M. Fujiwara, "Ueber die Wurzeln der Algebraischen Gleichungen", Tohoku Math. J, 8, pp.78~85 (1915)
  2. S. B. Kelleher, "Des Limites des zerosdun Polynome", J. Math. Pure Appl, 2, pp.169~171 (1916)
  3. M. Kuniyeda, "Note on the roots of algebraic equations", Tohoku Math. J, 9, pp.167~173 (1916)
  4. R.D. Carmichael, "Note on the roots of algebraic equations", Bull. Amer. Math. Soc, 21, pp.14~22 (1914)
  5. A.L. Cauchy, "Exercises de mathematique", Oeuvres, 9, pp.122~ (1829)
  6. M. Marden, "Geometry of Polynomials" (1949), American Mathematical Society
  7. Q. G. Mohammad, "On the zeros of polynomial", Amer. Math. Monthly, 72, pp.35~38 (1965)
  8. Q. G. Mohammad, "Location of the zeros of polynomials", Amer. Math. Monthly, 74, pp.290~292 (1967)
  9. P. Montel, "Sur la limite superieure du module des racines dune equation algebrique", C. R. Soc. Sci. Varsovie, 24, pp.317~326 (1932)
  10. X. Yang, "Holder's inequality", Appl. Math. Lett, 16, pp.897~903 (2003)
  11. Z. Rubinstein, "Analytical methods in the study of zeros of polynomials", Pacific J. Math, 13, pp.237~249 (1963)
  12. N.A. Rather, "Generalizations of Enestrom-Kakeya theorem and its extensions to analytic functions", J. Class. Anal, 16, pp.37~44 (2020)
  13. N.A. Rather, "On the regions containing all the zeros of polynomials and related analytic functions", Vest. Saint Petersburg Univ. Math. Mech. Astron, 8, pp.331~337 (2021)
  14. N.A Rather, "On the Enestrom-Kakeya theorem", Appl. Math. E-Notes, 22, pp.660~667 (2022)

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