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analysis.complex.schwarz

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

theorem complex.schwarz_aux {R₁ R₂ : } {c z : } {f : } (hd : R₁)) (h_maps : R₁) (metric.ball (f c) R₂)) (hz : z R₁) :
c z R₂ / R₁

An auxiliary lemma for complex.norm_dslope_le_div_of_maps_to_ball.

theorem complex.norm_dslope_le_div_of_maps_to_ball {E : Type u_1} [ E] {R₁ R₂ : } {f : E} {c z : } (hd : R₁)) (h_maps : R₁) (metric.ball (f c) R₂)) (hz : z R₁) :
c z R₂ / R₁

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

theorem complex.affine_of_maps_to_ball_of_exists_norm_dslope_eq_div {E : Type u_1} [ E] {R₁ R₂ : } {f : E} {c z₀ : } (hd : R₁)) (h_maps : R₁) (metric.ball (f c) R₂)) (h_z₀ : z₀ R₁) (h_eq : c z₀ = R₂ / R₁) :
(λ (z : ), f c + (z - c) c z₀) 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.

theorem complex.affine_of_maps_to_ball_of_exists_norm_dslope_eq_div' {E : Type u_1} [ E] {R₁ R₂ : } {f : E} {c : } (hd : R₁)) (h_maps : R₁) (metric.ball (f c) R₂)) (h_z₀ : (z₀ : ) (H : z₀ R₁), c z₀ = R₂ / R₁) :
(C : E), C = R₂ / R₁ (λ (z : ), f c + (z - c) C) R₁)
theorem complex.norm_deriv_le_div_of_maps_to_ball {E : Type u_1} [ E] {R₁ R₂ : } {f : E} {c : } (hd : R₁)) (h_maps : R₁) (metric.ball (f c) R₂)) (h₀ : 0 < R₁) :
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.

theorem complex.dist_le_div_mul_dist_of_maps_to_ball {E : Type u_1} [ E] {R₁ R₂ : } {f : E} {c z : } (hd : R₁)) (h_maps : R₁) (metric.ball (f c) R₂)) (hz : z R₁) :
has_dist.dist (f z) (f c) R₂ / R₁ *

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 complex.abs_deriv_le_div_of_maps_to_ball {f : } {c : } {R₁ R₂ : } (hd : R₁)) (h_maps : 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.

theorem complex.abs_deriv_le_one_of_maps_to_ball {f : } {c : } {R : } (hd : R)) (h_maps : R) R)) (hc : f c = c) (h₀ : 0 < R) :

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 complex.dist_le_dist_of_maps_to_ball_self {f : } {c z : } {R : } (hd : R)) (h_maps : R) R)) (hc : f c = c) (hz : z R) :

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 complex.abs_le_abs_of_maps_to_ball_self {f : } {z : } {R : } (hd : R)) (h_maps : R) R)) (h₀ : f 0 = 0) (hz : < R) :

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.