Theorems · Theorem · complex analysis

Complex.norm_le_norm_of_mapsTo_ball_self

Deprecated since 2026-01-03Use Complex.norm_le_norm_of_mapsTo_ball instead.

∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] [inst_2 : NormedAddCommGroup F]
  [inst_3 : NormedSpace ℂ F] {R : ℝ} {f : E → F} {z : E},
  DifferentiableOn ℂ f (Metric.ball 0 R) →
    Set.MapsTo f (Metric.ball 0 R) (Metric.closedBall 0 R) → f 0 = 0 → ‖z‖ < R → ‖f z‖ ≤ ‖z‖

Alias of Complex.norm_le_norm_of_mapsTo_ball. The Schwarz Lemma. Let f : E → F be a complex analytic on an open ball with center at the origin. If f sends this ball to the closed ball with center 0 of the same radius and f 0 = 0, then for any point z of this disk we have ‖f z‖ ≤ ‖z‖.

Defined in

Mathlib.Analysis.Complex.Schwarz

Cited by

0 results in Mathlib

Foundations

Depth 296 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace