International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 2, Pages 289-296

A generalization of an inequality of Zygmund

R. Peretz

Department of Mathematics, University of Michigan, Ann Arbor 48109, MI, USA

Received 17 October 1990

Copyright © 1993 R. Peretz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 maxzD|p(z)|nRmaxzD|p(z)| with equality iff p(z)=AZn. However it is true that we have the following better inequallty: maxzD|p(z)|nRmaxzD|Rep(z)| 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 1p< we have {02π|p(eix)|pdx}1/pApn{02π|Rep(eix)|pdx}1/p where App=π1/2Γ(12p+1)Γ(12p+12) 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 g(eix) is differentiable and if p(z) is a polynomial of degree n then for any 1p< we have {02π|g(eiθ)p(g(eiθ))|pdθ}1/pApnmaxβ{02π|Re{p(eiβg(eiθ))}|pdθ}1/p with equality iff p(z)=Azn.

We then obtain some conclusions for Schlicht Functions.