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
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.
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.