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)=z^n-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).
- Publication:
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arXiv e-prints
- Pub Date:
- August 2008
- DOI:
- 10.48550/arXiv.0808.0717
- arXiv:
- arXiv:0808.0717
- Bibcode:
- 2008arXiv0808.0717F
- Keywords:
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- Mathematics - Classical Analysis and ODEs;
- Mathematics - Complex Variables;
- 30C10
- E-Print:
- 18 pages