Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T19:29:18.925Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Problem Related to The Conjecture of Sendov About The Critical Points of a Polynomial

Published online by Cambridge University Press:  20 November 2018

Q. I. Rahman  and
Q. M. Tariq
Q. I. Rahman
Affiliation:
Département de Mathématiques et de Statistique Université de Montréal Montréal, PQ H3C 3J7
Q. M. Tariq
Affiliation:
Département de Mathématiques et de Statistique Université de Montréal Montréal, PQ H3C 3J7
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let P be a polynomial of degree n having all its zeros in the closed unit disk. Given that a is a zero (of P) of multiplicity k we seek to determine the radius ρ(n; k; a) of the smallest disk centred at a containing at least k zeros of the derivative P'. In the case k = 1 the answer has been conjectured to be 1 and is known to be true for n ≦ 5. We prove that ρ(n; k; a) ≦ 2k/(k + 1) for arbitrary k ∊ N and n ≦ k + 4.

Keywords

30C15 (Primary)
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 4 , 01 December 1987 , pp. 476 - 480
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Bojanov, B.D., Rahman, Q.I. and Szynal, J., On a conjecture of Sendov about the critical points of a polynomial, Math. Z. 190 (1985), pp. 281–285.Google Scholar
2. Dieudonné, J., La théorie analytique des polynômes d'une variable, Mémor. Sci. Math. No. 93 (1938).Google Scholar
3. Goodman, A.W., Rahman, Q.I. and Ratti, J.S., On the zeros of a polynomial and its derivative, Proc. Amer. Math. Soc. 21 (1969), pp. 273–274.Google Scholar
4. Joyal, A., On the zeros of a polynomial and its derivative, J. Math. Anal. Appl. 26 (1969), pp. 315–317.Google Scholar
5. Marden, M., Geometry of polynomials, Math. Surveys no. 3 , American Mathematical Society, 1966.Google Scholar
6. Marden, M., Conjectures on the critical points of a polynomial, Amer. Math. Monthly 90 (1983), pp. 267–276.Google Scholar
7. Meir, A. and Sharma, A., On Ilyeffs conjecture, Pacific J. Math. 31 (1969), pp. 459–467.Google Scholar
8. Schmeisser, G., Bemerkungen zu einer Vermutung von Ilieff, Math. Z. III (1969), pp. 121–125.Google Scholar
9. Szegö, G., Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen, Math. Z. 13 (1922), pp. 28–55.Google Scholar
10. Tariq, Q.M., On the zeros of a polynomial and its derivative, J. Univ. Kuwait (Sci.) 13 (1986), pp. 17–19.Google Scholar
11. Tariq, Q.M., On the zeros of a polynomial and its derivative. II, J. Univ. Kuwait (Sci.) 13 (1986), pp. 151–155.Google Scholar