On the length of lemniscates

Alexandre Eremenko; Walter Hayman

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arXiv · 0805.2295

On the length of lemniscates

Alexandre Eremenko·Walter Hayman

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Abstract

We show that for a monic polynomial p of degree d, the length of the level set {z: |p(z)|=1} is at most 9.2 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 this level set is connected.

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Alexandre Eremenko, Walter Hayman. 2008-05-15. On the length of lemniscates. https://arxiv.org/abs/0805.2295

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