A question concerning a polynomial inequality and an answer

Let M 2 ( g ; ρ ) denote the L 2 mean of g on the circle | z | = ρ . We prove that for any polynomial f ( z ) ≔ ∑ k = 0 n a k z k of degree at most n , with | a n − k | = | a k | for k = 0 , 1 , … , n , the ratio M 2 ( f ′ ; ρ ) / M 2 ( f ; 1 ) is maximized by f ( z ) ≔ 1 + z n for all ρ ∈ [ 2 − 1 / n , ∞ ) . At least in the case where n is even, the restriction on ρ cannot be relaxed.