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

  • Q. I. Rahman, Q. M. Tariq
  • Journal:
    Canadian Mathematical Bulletin / Volume 30 / Issue 4 / 01 December 1987
    Published online by Cambridge University Press:
    20 November 2018, pp. 476-480
    Print publication:
    01 December 1987
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  • 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.