New estimates for the length of the Erdos-Herzog-Piranian lemniscate

Details New estimates for the length of the Erdos-Herzog-Piranian lemniscate New estimates for the length of the Erdos-Herzog-Piranian lemniscate

2008-08-05

arxiv.org/abs/0808.0717v1

Mathematics

Classical Analysis and ODEs

18 pages

Scientific paper

Let p(z) be a monic polynomial of degree n. Consider the lemniscate
L={z:|p(z)|=1}. It has been conjectured that L has the largest length when
p(z)=z^n-1. We show that the length of L attains a local maximum at this
polynomial and prove the asymptotically sharp bound |L|<2n+o(n).

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