1. FormalConjectures
  2. ErdosProblems
  3. «513»
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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 Realopen Filter Polynomialnamespace Erdos513noncomputable 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 erdos_513 : answer(sorry) = ⨆ f : {f : ℂ → ℂ // Transcendental ℂ[X] f ∧ Differentiable ℂ f}, (liminf (fun r : ℝ => ratio r f) atTop) := ⊢ sorry = ⨆ f, liminf (fun r ↦ ratio r ↑f) atTop All 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 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 := ⊢ ∃ c > 0, ⨆ f, liminf (fun r ↦ ratio r ↑f) atTop ≤ 2 / π - c All goals completed! 🐙

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

@[category research solved, AMS 30] theorem 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 := ⊢ ⨆ f, liminf (fun r ↦ ratio r ↑f) atTop > 1 / 2 All goals completed! 🐙end Erdos513