International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 2, Pages 289-296
doi:10.1155/S0161171293000341

A generalization of an inequality of Zygmund

Abstract

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 maxz∈D|p′(z)|≤nRmaxz∈D|p(z)| with equality iff p(z)=AZn. However it is true that we have the following better inequallty: maxz∈D|p′(z)|≤nRmaxz∈D|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 1≤p<∞ we have {∫02π|p′(eix)|pdx}1/p≤Apn{∫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 1≤p<∞ we have {∫02π|g(eiθ)p′(g(eiθ))|pdθ}1/p≤Apnmaxβ{∫02π|Re{p(eiβg(eiθ))}|pdθ}1/p with equality iff p(z)=Azn.We then obtain some conclusions for Schlicht Functions.