An Lp inequality for polynomials

Let Pn be the class of all polynomials of degree at most n, and letMp(g;ρ) denote the Lp mean of g on the circle of radius ρ centered at the origin. We specify a number ρ∗ ∈ (0, 1), depending on n and k, such that for any f ∈ Pn, the ratio Mp(f (k);ρ)/Mp(f ;1) is maximized by f (z) := zn for all ρ ∈ [ρ∗,∞) and p 1. The interest of the result lies in the fact that ρ∗ is strictly less than 1. © 2007 Elsevier Inc. All rights reserved