| import OAI.Analysis.IntegralMeans.SharpModel |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory |
| open scoped ENNReal |
| namespace Brennan.Sharp |
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| lemma koebeInverse_left_inverse {z : ℂ} (hz : z ∈ disk) : |
| koebeInverse (koebeMap z) = z := |
| koebeMap_schlicht.1.2.leftInvOn_invFunOn hz |
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| lemma koebeInverse_mapsTo : MapsTo koebeInverse koebeDomain disk := by |
| rintro _ ⟨z, hz, rfl⟩ |
| rwa [koebeInverse_left_inverse hz] |
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| lemma koebeInverse_right_inverse {z : ℂ} (hz : z ∈ koebeDomain) : |
| koebeMap (koebeInverse z) = z := by |
| rcases hz with ⟨w, hw, rfl⟩ |
| rw [koebeInverse_left_inverse hw] |
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| lemma koebe_deriv_norm_left {w : ℂ} (hz : -1 + w ∈ disk) : |
| ‖deriv koebeMap (-1 + w)‖ = ‖w‖ / ‖2 - w‖ ^ 3 := by |
| rw [(koebeMap_hasDerivAt hz).deriv] |
| have hnum : 1 + (-1 + w) = w := by ring |
| have hden : 1 - (-1 + w) = 2 - w := by ring |
| rw [hnum, hden, norm_div, norm_pow] |
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| lemma koebe_deriv_norm_right {w : ℂ} (hz : 1 - w ∈ disk) : |
| ‖deriv koebeMap (1 - w)‖ = ‖2 - w‖ / ‖w‖ ^ 3 := by |
| rw [(koebeMap_hasDerivAt hz).deriv] |
| have hnum : 1 + (1 - w) = 2 - w := by ring |
| have hden : 1 - (1 - w) = w := by ring |
| rw [hnum, hden, norm_div, norm_pow] |
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| lemma one_le_norm_two_sub {w : ℂ} (hw : ‖w‖ < 1 / 2) : |
| 1 ≤ ‖2 - w‖ := by |
| have h := norm_sub_norm_le (2 : ℂ) w |
| norm_num at h |
| linarith |
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| lemma koebe_negative_endpoint_lower {w : ℂ} |
| (hz : -1 + w ∈ disk) (hw : ‖w‖ < 1 / 2) : |
| ‖w‖ ^ (-2 : ℝ) ≤ ‖deriv koebeMap (-1 + w)‖ ^ (-2 : ℝ) := by |
| have hd : 0 < ‖deriv koebeMap (-1 + w)‖ := |
| norm_pos_iff.mpr (univalent_deriv_ne_zero Metric.isOpen_ball koebeMap_schlicht.1 hz) |
| have hb : ‖deriv koebeMap (-1 + w)‖ ≤ ‖w‖ := by |
| rw [koebe_deriv_norm_left hz] |
| exact div_le_self (norm_nonneg _) (one_le_pow₀ (one_le_norm_two_sub hw)) |
| exact Real.rpow_le_rpow_of_nonpos hd hb (by norm_num) |
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| lemma koebe_positive_endpoint_lower {w : ℂ} |
| (hz : 1 - w ∈ disk) (h0 : 0 < ‖w‖) (hw : ‖w‖ < 1 / 2) : |
| ‖w‖ ^ (-2 : ℝ) ≤ ‖deriv koebeMap (1 - w)‖ ^ (2 / 3 : ℝ) := by |
| have hb : 1 / ‖w‖ ^ 3 ≤ ‖deriv koebeMap (1 - w)‖ := by |
| rw [koebe_deriv_norm_right hz] |
| exact div_le_div_of_nonneg_right (one_le_norm_two_sub hw) (by positivity) |
| have hp := Real.rpow_le_rpow (by positivity : 0 ≤ 1 / ‖w‖ ^ 3) hb |
| (by norm_num : (0 : ℝ) ≤ 2 / 3) |
| have he : (1 / ‖w‖ ^ 3) ^ (2 / 3 : ℝ) = ‖w‖ ^ (-2 : ℝ) := by |
| rw [one_div, Real.inv_rpow (by positivity), ← Real.rpow_natCast ‖w‖ 3, |
| ← Real.rpow_mul h0.le] |
| norm_num [Real.rpow_neg, Real.rpow_two] |
| rwa [he] at hp |
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| end Brennan.Sharp |
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| end |
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| end OAI |
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