A Generalization of an Inequality of Zygmund

The well known Bernstein inequality 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 [formula omitted] with equality iff p(z) = AZⁿ. However it is true that we have the following better inequality: [formula omitted] with equality iff p(z) = AZⁿ.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 [formula omitted] where [formula omitted] with equality iff p(z) =AZⁿ In this note we generalize the last result to domains different from Euclidean disks by showing the following: If geeix) is differentiable and if p(z) is a polynomial of degree n then for any 1 ≤ P < ∞ [formula omitted]we have with equality iff p(z) = Azⁿ. We then obtain some conclusions for Schlicht Functions.