Schwarz lemma #

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In this file we prove several versions of the Schwarz lemma.

complex.norm_deriv_le_div_of_maps_to_ball, complex.abs_deriv_le_div_of_maps_to_ball: 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;
complex.dist_le_div_mul_dist_of_maps_to_ball: if f : ℂ → E sends an open disk with center c and radius R₁ to an open disk 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;
complex.abs_deriv_le_one_of_maps_to_ball: 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;
complex.dist_le_dist_of_maps_to_ball: 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;
complex.abs_le_abs_of_maps_to_ball: 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.

Implementation notes #

We prove some versions of the Schwarz lemma for a map f : ℂ → E taking values in any normed space over complex numbers.

TODO #

Prove that these inequalities are strict unless f is an affine map.
Prove that any diffeomorphism of the unit disk to itself is a Möbius map.

Tags #

Schwarz lemma

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theorem complex.schwarz_aux {R₁ R₂ : ℝ} {c z : ℂ} {f : ℂ → ℂ} (hd : differentiable_on ℂ f (metric.ball c R₁)) (h_maps : set.maps_to f (metric.ball c R₁) (metric.ball (f c) R₂)) (hz : z ∈ metric.ball c R₁) :

‖dslope f c z‖ ≤ R₂ / R₁

An auxiliary lemma for complex.norm_dslope_le_div_of_maps_to_ball.

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theorem complex.norm_dslope_le_div_of_maps_to_ball {E : Type u_1} [normed_add_comm_group E] [normed_space ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c z : ℂ} (hd : differentiable_on ℂ f (metric.ball c R₁)) (h_maps : set.maps_to f (metric.ball c R₁) (metric.ball (f c) R₂)) (hz : z ∈ metric.ball c R₁) :

‖dslope f c z‖ ≤ R₂ / R₁

Two cases of the Schwarz Lemma (derivative and distance), merged together.

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theorem complex.affine_of_maps_to_ball_of_exists_norm_dslope_eq_div {E : Type u_1} [normed_add_comm_group E] [normed_space ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c z₀ : ℂ} [complete_space E] [strict_convex_space ℝ E] (hd : differentiable_on ℂ f (metric.ball c R₁)) (h_maps : set.maps_to f (metric.ball c R₁) (metric.ball (f c) R₂)) (h_z₀ : z₀ ∈ metric.ball c R₁) (h_eq : ‖dslope f c z₀‖ = R₂ / R₁) :

set.eq_on f (λ (z : ℂ), f c + (z - c) • dslope f c z₀) (metric.ball c R₁)

Equality case in the Schwarz Lemma: in the setup of norm_dslope_le_div_of_maps_to_ball, if ‖dslope f c z₀‖ = R₂ / R₁ holds at a point in the ball then the map f is affine.

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theorem complex.affine_of_maps_to_ball_of_exists_norm_dslope_eq_div' {E : Type u_1} [normed_add_comm_group E] [normed_space ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c : ℂ} [complete_space E] [strict_convex_space ℝ E] (hd : differentiable_on ℂ f (metric.ball c R₁)) (h_maps : set.maps_to f (metric.ball c R₁) (metric.ball (f c) R₂)) (h_z₀ : ∃ (z₀ : ℂ) (H : z₀ ∈ metric.ball c R₁), ‖dslope f c z₀‖ = R₂ / R₁) :

∃ (C : E), ‖C‖ = R₂ / R₁ ∧ set.eq_on f (λ (z : ℂ), f c + (z - c) • C) (metric.ball c R₁)

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theorem complex.norm_deriv_le_div_of_maps_to_ball {E : Type u_1} [normed_add_comm_group E] [normed_space ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c : ℂ} (hd : differentiable_on ℂ f (metric.ball c R₁)) (h_maps : set.maps_to f (metric.ball c R₁) (metric.ball (f c) R₂)) (h₀ : 0 < R₁) :

‖deriv f c‖ ≤ R₂ / R₁

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.

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theorem complex.dist_le_div_mul_dist_of_maps_to_ball {E : Type u_1} [normed_add_comm_group E] [normed_space ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c z : ℂ} (hd : differentiable_on ℂ f (metric.ball c R₁)) (h_maps : set.maps_to f (metric.ball c R₁) (metric.ball (f c) R₂)) (hz : z ∈ metric.ball c R₁) :

has_dist.dist (f z) (f c) ≤ R₂ / R₁ * has_dist.dist z c

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.

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theorem complex.abs_deriv_le_div_of_maps_to_ball {f : ℂ → ℂ} {c : ℂ} {R₁ R₂ : ℝ} (hd : differentiable_on ℂ f (metric.ball c R₁)) (h_maps : set.maps_to f (metric.ball c R₁) (metric.ball (f c) R₂)) (h₀ : 0 < R₁) :

⇑complex.abs (deriv f c) ≤ R₂ / R₁

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.

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theorem complex.abs_deriv_le_one_of_maps_to_ball {f : ℂ → ℂ} {c : ℂ} {R : ℝ} (hd : differentiable_on ℂ f (metric.ball c R)) (h_maps : set.maps_to f (metric.ball c R) (metric.ball c R)) (hc : f c = c) (h₀ : 0 < R) :

⇑complex.abs (deriv f c) ≤ 1

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.

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theorem complex.dist_le_dist_of_maps_to_ball_self {f : ℂ → ℂ} {c z : ℂ} {R : ℝ} (hd : differentiable_on ℂ f (metric.ball c R)) (h_maps : set.maps_to f (metric.ball c R) (metric.ball c R)) (hc : f c = c) (hz : z ∈ metric.ball c R) :

has_dist.dist (f z) c ≤ has_dist.dist z c

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.

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theorem complex.abs_le_abs_of_maps_to_ball_self {f : ℂ → ℂ} {z : ℂ} {R : ℝ} (hd : differentiable_on ℂ f (metric.ball 0 R)) (h_maps : set.maps_to f (metric.ball 0 R) (metric.ball 0 R)) (h₀ : f 0 = 0) (hz : ⇑complex.abs z < R) :

⇑complex.abs (f z) ≤ ⇑complex.abs z

The Schwarz Lemma: if f : ℂ → ℂ sends an open disk with center 0 to itself, the for any point z of this disk we have abs (f z) ≤ abs z.