| import OAI.Analysis.IntegralMeans.Koebe |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory Filter Function |
| open scoped Topology |
| namespace Brennan |
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| 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 |
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| 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 |
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| 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⟩ |
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| 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) |
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| 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 |
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| 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⟩)) |
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| end Brennan |
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| end |
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| end OAI |
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