The arc length of the lemniscate {|𝑝(𝑧)|=1}

Borwein, Peter

doi:10.1090/S0002-9939-1995-1223265-3

The arc length of the lemniscate $\{\vert p(z)\vert =1\}$
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by Peter Borwein

Proc. Amer. Math. Soc. 123 (1995), 797-799

DOI: https://doi.org/10.1090/S0002-9939-1995-1223265-3

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Abstract:

We show that the length of the set \[ \left \{ {z \in \mathbb {C}:|\prod \limits _{i = 1}^n {(z - {\alpha _i})| = 1} } \right \}\] is at most $8\pi en$. This gives the correct rate of growth in a long-standing open problem of Erdös, Herzog, and Piranian and improves the previous bound of $74{n^2}$ due to Pommerenke.

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