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Erdős problem 1048

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If f∈C[x]f\in \mathbb{C}[x] is a monic polynomial with all roots satisfying ∣z∣≤r\lvert z\rvert \leq r for some r<2r<2, then must {z:∣f(z)∣<1}\{ z: \lvert f(z)\rvert <1\} have a connected component with diameter >2−r>2-r?

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8 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

1048.lean

Retained formal statement2 of 8

If 0≤r≤1/20\leq r\leq 1/2 then the component which contains 00 must have diameter ≥2\geq 2, which f(z)=znf(z)=z^n shows is best possible.

FormalConjectures/ErdosProblems/1048.leanErdos1048.erdos_1048.variants.diam_ge_two6 linesExact file
∀ (r : ℝ),  0 ≤ r →    r ≤ 1 / 2 →      ∀ (f : Polynomial ℂ),        f.Monic →          f.degree ≥ 1 → (∀ z ∈ f.roots, ‖z‖ ≤ r) → 2 ≤ Metric.ediam (connectedComponentIn (Erdos1043.levelSet f) 0)
SolvedStatement only, no proof