FormalConjectures.ErdosProblems.«115»

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import FormalConjecturesUtil

Erdős Problem 115

References:

erdosproblems.com/115

[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.

[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.

[ErLe94] Erëmenko, A. and Lempert, L., An extremal problem for polynomials. Proc. Amer. Math. Soc. (1994), 191-193.

[Po59a] Pommerenke, Ch., On the derivative of a polynomial. Michigan Math. J. (1959), 373-375.

open Filternamespace Erdos115

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.

@[category research solved, AMS 30, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos115.lean"]
theorem erdos_115 : answer(True) ↔
    ∀ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop, ∀ p : Polynomial ℂ, p.Monic → p.natDegree = n →
      IsConnected {z : ℂ | ‖p.eval z‖ ≤ 1} → ∀ z : ℂ, ‖p.eval z‖ ≤ 1 →
        ‖p.derivative.eval z‖ ≤ (1 / 2 + ε) * (n : ℝ) ^ 2 := by⊢ True ↔
  ∀ ε > 0,
    ∀ᶠ (n : ℕ) in atTop,
      ∀ (p : Polynomial ℂ),
        p.Monic →
          p.natDegree = n →
            IsConnected {z | ‖Polynomial.eval z p‖ ≤ 1} →
              ∀ (z : ℂ),
                ‖Polynomial.eval z p‖ ≤ 1 → ‖Polynomial.eval z (Polynomial.derivative p)‖ ≤ (1 / 2 + ε) * ↑n ^ 2
  sorryAll goals completed! 🐙

end Erdos115