New estimates for the length of the Erdos-Herzog-Piranian lemniscate
Abstract
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)=zn-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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