FormalConjectures.ErdosProblems.«513»

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import FormalConjecturesUtil

Erdős Problem 513

Reference:

erdosproblems.com/513

[ClHa64] Clunie, J. and Hayman, W. K., The maximum term of a power series. J. Analyse Math. (1964), 143-186.


open scoped Nat Real

open Filter Polynomial


namespace Erdos513


noncomputable def ratio (r : ℝ) (f : ℂ → ℂ) : ℝ :=
  (⨆ n, ‖iteratedDeriv n f 0 * (n ! : ℝ)⁻¹ * r ^ n‖) / (⨆ z : {z : ℂ // ‖z‖ = r}, ‖f z‖)

Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?

@[category research open, AMS 30]
theorem declaration uses 'sorry'erdos_513 : answer(sorry) =
    ⨆ f : {f : ℂ → ℂ // Transcendental ℂ[X] f ∧ Differentiable ℂ f},
    (liminf (fun r : ℝ => ratio r f) atTop) := by⊢ sorry = ⨆ f, liminf (fun r => ratio r ↑f) atTop
  sorryAll goals completed! 🐙

For all transcendental entire function f, liminf (fun r : ℝ => ratio r f) atTop ≤ 2 / π - c for some c > 0. This is proved in [ClHa64].

@[category research solved, AMS 30]
theorem declaration uses 'sorry'erdos_513.variants.upper_bound : ∃ c > 0,
    ⨆ f : {f : ℂ → ℂ // Transcendental ℂ[X] f ∧ Differentiable ℂ f},
    (liminf (fun r : ℝ => ratio r f) atTop) ≤ 2 / π - c := by⊢ ∃ c > 0, ⨆ f, liminf (fun r => ratio r ↑f) atTop ≤ 2 / π - c
  sorryAll goals completed! 🐙

For all transcendental entire function f, liminf (fun r : ℝ => ratio r f) atTop > 1 / 2.

@[category research solved, AMS 30]
theorem declaration uses 'sorry'erdos_513.variants.lower_bound :
    ⨆ f : {f : ℂ → ℂ // Transcendental ℂ[X] f ∧ Differentiable ℂ f},
    (liminf (fun r : ℝ => ratio r f) atTop) > 1 / 2 := by⊢ ⨆ f, liminf (fun r => ratio r ↑f) atTop > 1 / 2
  sorryAll goals completed! 🐙


end Erdos513