A question concerning a polynomial inequality and an answer Original Research Article

Let M2(g;ρ)M2(g;ρ) denote the L2L2 mean of gg on the circle |z|=ρ|z|=ρ. We prove that for any polynomial View the MathML sourcef(z)≔∑k=0nakzk of degree at most nn, with |an−k|=|ak||an−k|=|ak| for k=0,1,…,nk=0,1,…,n, the ratio M2(f′;ρ)/M2(f;1)M2(f′;ρ)/M2(f;1) is maximized by f(z)≔1+znf(z)≔1+zn for all ρ∈[2−1/n,∞)ρ∈[2−1/n,∞). At least in the case where nn is even, the restriction on ρρ cannot be relaxed.