A generalization of an inequality of Zygmund

The well known Bernstein Inequallty states that if D is a disk centered at the origin with radius R and if p ( z ) is a polynomial of degree n , then max z ∈ D | p ′ ( z ) | ≤ n R max z ∈ D | p ( z ) | with equality iff p ( z ) = A Z n . However it is true that we have the following better inequallty: max z ∈ D | p ′ ( z ) | ≤ n R max z ∈ D | Re p ( z ) | with equality iff p ( z ) = A Z n . This is a consequence of a general equality that appears in Zygmund [7] (and which is due to Bernstein and Szegö): For any polynomial p ( z ) of degree n and for any 1 ≤ p < ∞ we have { ∫ 0 2 π | p ′ ( e i x ) | p d x } 1 / p ≤ A p n { ∫ 0 2 π | Re p ( e i x ) | p d x } 1 / p where A p p = π 1 / 2 Γ ( 1 2 p + 1 ) Γ ( 1 2 p + 1 2 ) with equality iff p ( z ) = A Z n . In this note we generalize the last result to domains different from Euclidean disks by showing the following: If g ( e i x ) is differentiable and if p ( z ) is a polynomial of degree n then for any 1 ≤ p < ∞ we have { ∫ 0 2 π | g ( e i θ ) p ′ ( g ( e i θ ) ) | p d θ } 1 / p ≤ A p n max β { ∫ 0 2 π | Re { p ( e i β g ( e i θ ) ) } | p d θ } 1 / p with equality iff p ( z ) = A z n . We then obtain some conclusions for Schlicht Functions.