Journal of Inequalities and Applications
Volume 1 (1997), Issue 3, Pages 199-222

Small values of polynomials: Cartan, Pólya and others

D. S. Lubinsky

Department of Mathematics, Witwatersrand University, Wits, 2050, South Africa

Received 20 June 1996

Copyright © 1997 D. S. Lubinsky. 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.


Let P(z) be a monic polynomial of degree n, and α, ε>0. A classic lemma of Cartan asserts that the lemniscate E(P;ε):={z:|P(z)|εn} can be covered by balls Bj,1jn, whose diameters d(Bj) satisfy j=1p(d(Bj))αe(4ε)α. For a=2, this shows that E(P;ε) has an area at most πe(2ε)2. Pólya showed in this case that the sharp estimate is πε2. We discuss some of the ramifications of these estimates, as well as some of their close cousins, for example when P is normalized to have Lp norm 1 on some circle, and Remez’ inequality.