File: Mathlib/Analysis/Complex/Schwarz.lean

full_name: Complex.schwarz_aux
code: /-- An auxiliary lemma for `Complex.norm_dslope_le_div_of_mapsTo_ball`. -/
theorem schwarz_aux {f : ℂ → ℂ} (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
    ‖dslope f c z‖ ≤ R₂ / R₁ 
goal: ‖dslope f c z‖ ≤ R₂ / R₁

full_name: Complex.norm_dslope_le_div_of_mapsTo_ball
code: /-- Two cases of the **Schwarz Lemma** (derivative and distance), merged together. -/
theorem norm_dslope_le_div_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
    ‖dslope f c z‖ ≤ R₂ / R₁ 
goal: ‖dslope f c z‖ ≤ R₂ / R₁

full_name: Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div
code: /-- Equality case in the **Schwarz Lemma**: in the setup of `norm_dslope_le_div_of_mapsTo_ball`, if
`‖dslope f c z₀‖ = R₂ / R₁` holds at a point in the ball then the map `f` is affine. -/
theorem affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div [CompleteSpace E] [StrictConvexSpace ℝ E]
    (hd : DifferentiableOn ℂ f (ball c R₁)) (h_maps : Set.MapsTo f (ball c R₁) (ball (f c) R₂))
    (h_z₀ : z₀ ∈ ball c R₁) (h_eq : ‖dslope f c z₀‖ = R₂ / R₁) :
    Set.EqOn f (fun z => f c + (z - c) • dslope f c z₀) (ball c R₁) 
goal: Set.EqOn f (fun z => f c + (z - c) • dslope f c z₀) (Metric.ball c R₁)

full_name: Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div'
code: theorem affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div' [CompleteSpace E]
    [StrictConvexSpace ℝ E] (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : Set.MapsTo f (ball c R₁) (ball (f c) R₂))
    (h_z₀ : ∃ z₀ ∈ ball c R₁, ‖dslope f c z₀‖ = R₂ / R₁) :
    ∃ C : E, ‖C‖ = R₂ / R₁ ∧ Set.EqOn f (fun z => f c + (z - c) • C) (ball c R₁) 
goal: ∃ C, ‖C‖ = R₂ / R₁ ∧ Set.EqOn f (fun z => f c + (z - c) • C) (Metric.ball c R₁)

full_name: Complex.norm_deriv_le_div_of_mapsTo_ball
code: /-- The **Schwarz Lemma**: if `f : ℂ → E` sends an open disk with center `c` and a positive radius
`R₁` to an open ball with center `f c` and radius `R₂`, then the absolute value of the derivative of
`f` at `c` is at most the ratio `R₂ / R₁`. -/
theorem norm_deriv_le_div_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (h₀ : 0 < R₁) : ‖deriv f c‖ ≤ R₂ / R₁ 
goal: ‖deriv f c‖ ≤ R₂ / R₁

full_name: Complex.dist_le_div_mul_dist_of_mapsTo_ball
code: /-- The **Schwarz Lemma**: if `f : ℂ → E` sends an open disk with center `c` and radius `R₁` to an
open ball with center `f c` and radius `R₂`, then for any `z` in the former disk we have
`dist (f z) (f c) ≤ (R₂ / R₁) * dist z c`. -/
theorem dist_le_div_mul_dist_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
    dist (f z) (f c) ≤ R₂ / R₁ * dist z c 
goal: dist (f z) (f c) ≤ R₂ / R₁ * dist z c

full_name: Complex.abs_deriv_le_div_of_mapsTo_ball
code: /-- The **Schwarz Lemma**: if `f : ℂ → ℂ` sends an open disk with center `c` and a positive radius
`R₁` to an open disk with center `f c` and radius `R₂`, then the absolute value of the derivative of
`f` at `c` is at most the ratio `R₂ / R₁`. -/
theorem abs_deriv_le_div_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
    (h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (h₀ : 0 < R₁) : abs (deriv f c) ≤ R₂ / R₁ 
goal: Complex.abs (deriv f c) ≤ R₂ / R₁

full_name: Complex.abs_deriv_le_one_of_mapsTo_ball
code: /-- The **Schwarz Lemma**: if `f : ℂ → ℂ` sends an open disk of positive radius to itself and the
center of this disk to itself, then the absolute value of the derivative of `f` at the center of
this disk is at most `1`. -/
theorem abs_deriv_le_one_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R))
    (h_maps : MapsTo f (ball c R) (ball c R)) (hc : f c = c) (h₀ : 0 < R) : abs (deriv f c) ≤ 1 
goal: Complex.abs (deriv f c) ≤ 1

full_name: Complex.dist_le_dist_of_mapsTo_ball_self
code: /-- The **Schwarz Lemma**: if `f : ℂ → ℂ` sends an open disk to itself and the center `c` of this
disk to itself, then for any point `z` of this disk we have `dist (f z) c ≤ dist z c`. -/
theorem dist_le_dist_of_mapsTo_ball_self (hd : DifferentiableOn ℂ f (ball c R))
    (h_maps : MapsTo f (ball c R) (ball c R)) (hc : f c = c) (hz : z ∈ ball c R) :
    dist (f z) c ≤ dist z c 
goal: dist (f z) c ≤ dist z c

full_name: Complex.abs_le_abs_of_mapsTo_ball_self
code: /-- The **Schwarz Lemma**: if `f : ℂ → ℂ` sends an open disk with center `0` to itself, then for any
point `z` of this disk we have `abs (f z) ≤ abs z`. -/
theorem abs_le_abs_of_mapsTo_ball_self (hd : DifferentiableOn ℂ f (ball 0 R))
    (h_maps : MapsTo f (ball 0 R) (ball 0 R)) (h₀ : f 0 = 0) (hz : abs z < R) :
    abs (f z) ≤ abs z 
goal: Complex.abs (f z) ≤ Complex.abs z