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ABSTRACT. Let Pn be the normed linear space of all polynomials p of degree < n such that p(l) = 0 and ||p|| = (f |p(x)|2 dx)1/2. We determine sharp upper bounds for |on|/||p|| and |an_i|/||p|| as p(x) := X^n=oa"x" va"es mPn-

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.