The board / polynomials / #115
Erdős Problem #115
PROVED LEANdatabase updated 2026-03-03
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
If \(p(z)\) is a polynomial of degree \(n\) such that \(\{z : \lvert p(z)\rvert\leq 1\}\) is connected then is it true that \[\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?\] Eremenko and Lempert [ErLe94] have shown this is true, and in fact Chebyshev polynomials are the extreme examples.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/115.Prizenone
Topicspolynomials, analysis
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/115
Attempts
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