SMALE'S PROBLEM FOR CRITICAL POINTS ON CERTAIN TWO RAYS.
- Resource Type
- Article
- Authors
- Hinkkanen, Aimo; Kayumov, Ilgiz
- Source
- Journal of the Australian Mathematical Society. Apr2010, Vol. 88 Issue 2, p183-191. 9p.
- Subject
- *POLYNOMIALS
*CRITICAL point theory
*MATHEMATICAL functions
*MATHEMATICAL analysis
*GEOMETRIC function theory
*COMPLEX variables
*MATHEMATICAL inequalities
*INFINITE processes
*ALGEBRA
- Language
- ISSN
- 1446-7887
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 ξ of f with ∣ f (ξ)/ ξ∣ ≤ 1/2 provided that the critical points of f lie in the sector {reiθ : r > 0; ∣θ∣ ≤ π/6}, and ∣ f (ξ)/ξ∣ < 2/3 if they lie in the union of the two rays {1 + re±iθ : r ≥ 0}, where 0 < θ ≤ π/2. [ABSTRACT FROM AUTHOR]