An Lp Inequality for a Polynomial and Its Derivative

Let P(z) = an Πnν=1 (z − zν), an ≠ 0 be a polynomial of degree n. It is known that if |zν| ≥ Kν ≥ 1, 1 ≤ ν ≤ n, then for p ≥ 1,[formula] where [formula] and [formula] This inequality is best possible in the case Kν = 1, 1 ≤ ν ≤ n, and equality holds for the polynomial (z + 1)n. In this paper, we extend the above inequality to values of p ∈ [0, 1) and thus conclude that this inequality in fact holds for all p ≥ 0.