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  • SMALE’S PROBLEM FOR CRITICAL POINTS ON CERTAIN TWO RAYS

  • Part of
  • AIMO HINKKANEN, ILGIZ KAYUMOV
  • Journal:
    Journal of the Australian Mathematical Society / Volume 88 / Issue 2 / April 2010
    Published online by Cambridge University Press:
    25 March 2010, pp. 183-191
    Print publication:
    April 2010
    • Article
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  • Let f be a polynomial of degree n≥2 with f(0)=0 and f′(0)=1. We prove that there is a critical point ζ of f with ∣f(ζ)/ζ∣≤1/2 provided that the critical points of f lie in the sector {reiθ:r>0,∣θ∣≤π/6}, and ∣f(ζ)/ζ∣<2/3 if they lie in the union of the two rays {1+re±iθ:r≥0}, where 0<θ≤π/2.