An Lp inequality for polynomials

Let Pn be the class of all polynomials of degree at most n, and let Mp(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.