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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | import OAI.Analysis.IntegralMeans.Koebe
namespace OAI
noncomputable section
open Set MeasureTheory Filter Function
open scoped Topology
namespace Brennan
lemma integralMean_pos {f : ℂ → ℂ} (hf : UnivalentOn f disk) {r : ℝ}
(hr0 : 0 ≤ r) (hr1 : r < 1) (t : ℝ) : 0 < integralMean f t r := by
apply mul_pos (by positivity)
apply intervalIntegral.integral_pos (by linarith [Real.pi_pos])
(continuous_circle_deriv_rpow hf hr0 hr1 t).continuousOn
(fun θ _ => Real.rpow_nonneg (norm_nonneg _) t)
refine ⟨0,⟨by linarith [Real.pi_pos],by linarith [Real.pi_pos]⟩,?_⟩
exact Real.rpow_pos_of_pos (norm_pos_iff.mpr (univalent_deriv_ne_zero Metric.isOpen_ball hf
(circlePoint_mem_disk hr0 hr1 0))) t
lemma tendsto_log_boundary : Tendsto (fun r : ℝ => Real.log (1/(1-r))) (𝓝[<] (1:ℝ)) atTop := by
have hδ : Tendsto (fun r : ℝ => 1-r) (𝓝[<] (1:ℝ)) (𝓝[>] (0:ℝ)) := by
apply tendsto_nhdsWithin_iff.mpr
constructor
· have hc : Continuous (fun r : ℝ => 1-r) := continuous_const.sub continuous_id
have h := hc.continuousAt.tendsto.mono_left
(nhdsWithin_le_nhds : 𝓝[<] (1:ℝ) ≤ 𝓝 1)
simpa only [sub_self] using h
· filter_upwards [self_mem_nhdsWithin] with r hr
exact sub_pos.mpr (show r < 1 from hr)
have h := tendsto_neg_atBot_atTop.comp (Real.tendsto_log_nhdsGT_zero.comp hδ)
simpa only [Function.comp_def,one_div,Real.log_inv] using h
lemma eventually_radius : ∀ᶠ r : ℝ in 𝓝[<] (1:ℝ), 1/2 ≤ r ∧ r < 1 := by
filter_upwards [eventually_nhdsWithin_of_eventually_nhds (eventually_gt_nhds (by norm_num : (1/2:ℝ) < 1)),self_mem_nhdsWithin] with r hr hr1
exact ⟨hr.le,hr1⟩
lemma log_power_upper {M C a r : ℝ} (hM : 0 < M) (hC : 0 < C)
(hr : 0 < r) (hr1 : r < 1) (h : M ≤ C*(1-r)^(-a)) :
Real.log M / Real.log (1/(1-r)) ≤ Real.log C / Real.log (1/(1-r)) + a := by
have hd : 0 < 1-r := by linarith
have hl : 0 < Real.log (1/(1-r)) := Real.log_pos ((one_lt_div hd).mpr (by linarith))
have hb := Real.log_le_log hM h
rw [Real.log_mul hC.ne' (Real.rpow_pos_of_pos hd _).ne',Real.log_rpow hd] at hb
have he : Real.log (1/(1-r)) = -Real.log (1-r) := by simp [one_div]
rw [div_le_iff₀ hl]
calc
Real.log M ≤ Real.log C + (-a)*Real.log (1-r) := hb
_ = (Real.log C / Real.log (1/(1-r))+a)*Real.log (1/(1-r)) := by
rw [add_mul,div_mul_cancel₀ _ hl.ne',he]; ring
lemma beta_inverse_le_one {f : ℂ → ℂ} (hf : Schlicht f) : beta f (-2) ≤ 1 := by
have hb (ε : ℝ) (hε : 0 < ε) : beta f (-2) ≤ ((1+ε:ℝ):EReal) := by
obtain ⟨C,hC,hbound⟩ := uniform_inverse_square_bound ε hε
have hc : Tendsto (fun r : ℝ => ((Real.log C/Real.log (1/(1-r))+(1+ε):ℝ):EReal))
(𝓝[<] (1:ℝ)) (𝓝 ((1+ε:ℝ):EReal)) := by
have hh := (tendsto_log_boundary.const_div_atTop (Real.log C)).add_const (1+ε)
simpa [Function.comp_def] using continuous_coe_real_ereal.continuousAt.tendsto.comp hh
rw [← hc.limsup_eq]
refine limsup_le_limsup ?_ (by isBoundedDefault) (by isBoundedDefault)
filter_upwards [eventually_radius] with r hr
apply EReal.coe_le_coe
exact log_power_upper (integralMean_pos hf.1 (by linarith [hr.1]) hr.2 (-2)) hC
(by linarith [hr.1]) hr.2 (by simpa only [show -(1+ε) = -1-ε by ring] using hbound f hf r hr.1 hr.2)
by_contra h
obtain ⟨a,ha,hfa⟩ := EReal.exists_between_coe_real (lt_of_not_ge h)
have ha' : 1 < a := by exact_mod_cast ha
have he := hb ((a-1)/2) (by linarith)
have he' : ((1+(a-1)/2:ℝ):EReal) < a := EReal.coe_lt_coe_iff.mpr (by linarith)
exact (not_lt_of_ge he) (he'.trans hfa)
lemma koebe_beta_ge_one : 1 ≤ beta koebeMap (-2) := by
let C : ℝ := 1/(4*Real.pi)
have hC : 0 < C := by dsimp [C]; positivity
have hc : Tendsto (fun r : ℝ => ((Real.log C/Real.log (1/(1-r))+1:ℝ):EReal))
(𝓝[<] (1:ℝ)) (𝓝 (1:EReal)) := by
have hh := (tendsto_log_boundary.const_div_atTop (Real.log C)).add_const 1
simpa [Function.comp_def] using continuous_coe_real_ereal.continuousAt.tendsto.comp hh
rw [← hc.limsup_eq]
refine limsup_le_limsup ?_ (by isBoundedDefault) (by isBoundedDefault)
filter_upwards [eventually_radius] with r hr
apply EReal.coe_le_coe
have hd : 0 < 1-r := by linarith [hr.2]
have hl : 0 < Real.log (1/(1-r)) := Real.log_pos ((one_lt_div hd).mpr (by linarith [hr.1]))
have hb := koebe_mean_lower hr.1 hr.2
have hid : 1/(4*Real.pi*(1-r)) = C*(1/(1-r)) := by dsimp [C]; field_simp
rw [hid] at hb
have hh := Real.log_le_log (mul_pos hC (by positivity)) hb
rw [Real.log_mul hC.ne' (by positivity)] at hh
rw [le_div_iff₀ hl,add_mul,div_mul_cancel₀ _ hl.ne',one_mul]
exact hh
lemma inverse_spectrum_eq_one : spectrum (-2) = 1 := by
apply le_antisymm
· apply sSup_le
rintro _ ⟨f,hf,rfl⟩
exact beta_inverse_le_one hf
· exact koebe_beta_ge_one.trans (le_sSup (show beta koebeMap (-2) ∈ {b | ∃ f,Schlicht f ∧ beta f (-2) = b} from
⟨koebeMap,koebeMap_schlicht,rfl⟩))
end Brennan
end
end OAI
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