On the Length of Lemniscates

摘要

For a monic polynomial p of degree d, we write E(p) := { z : |p(z)| = l }. A conjecture of Erdos, Herzog and Piranian [4], repeated by Erdos in [5, Prob. 4. 10] and elsewhere, is that the length E(p) 1 is maximal when p(z) = zd + l. It is easy to see that, in this conjectured extremal case,E(p)1 = Zd + O(l) when d → ∞. The first upper estimate E(p)|≤ 74d2 is due to Pommerenke [10J. Recently, Borwein [2] gave an estimate that is linear in d, namely E(p)|≤8πed ≈ 68.32d. Here we improve Borwein's result.