On the length of lemniscates.

Abstract

We show that for a polynomial p(z) = z + . . . the length of the level set E(p) := {z : |p(z)| = 1} is at most 9.173 d, which improves an earlier estimate due to P. Borwein. For d = 2 we show that the extremal level set is the Bernoullis’ Lemniscate. One ingredient of our proofs is the fact that for an extremal polynomial the set E(p) is connected. For a monic polynomial p of degree d we write E(p) := {z : |p(z)| = 1}. A conjecture of Erdős, Herzog and Piranian [4], repeated by Erdős in [5] and elsewhere, is that the length |E(p)| is maximal when p(z) := z + 1. It is easy to see that in this conjectured extremal case |E(p)| = 2d+O(1) when d→∞. The first upper estimate |E(p)| ≤ 74d2 is due to Pommerenke [8]. Recently P. Borwein [2] gave an estimate which is linear in d, namely |E(p)| ≤ 8πed ≈ 68.32d. Here we improve Borwein’s result. Let α0 be the least upper bound of perimeters of the convex hulls of compact connected sets of logarithmic capacity 1. The precise value of α0 is not known, but Pommerenke [10] proved the estimate α0 < 9.173. The conjectured value is α0 = 3 3/222/3 ≈ 8.24. Theorem 1 For monic polynomials p of degree d |E(p)| ≤ α0d < 9.173d. A similar problem for rational functions turns out to be much easier, and can be solved completely by means of Lemma 1 below. ∗Supported by EPSRC grant GR/L 35546 at Imperial College and by NSF grant DMS9800084