Theorems · Theorem · complex analysis

Complex.dist_le_div_mul_dist_of_mapsTo_ball

∀ {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] {R₁ R₂ : ℝ} {f : E → F} {c z : E},
  DifferentiableOn ℂ f (Metric.ball c R₁) →
    Set.MapsTo f (Metric.ball c R₁) (Metric.closedBall (f c) R₂) →
      z ∈ Metric.ball c R₁ → dist (f z) (f c) ≤ R₂ / R₁ * dist z c

The Schwarz Lemma. Let f : E → F be a complex analytic function on an open ball with center c and radius R₁. If f sends this ball to a closed ball with center f c and radius R₂, then for any z in the former ball we have dist (f z) (f c) ≤ (R₂ / R₁) * dist z c.

Defined in: Mathlib.Analysis.Complex.Schwarz
Cited by: 3 results in Mathlib
Foundations: Depth 293 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup NormedSpace NormedAddCommGroup NormedSpace

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Setstatement · cited by 53,352
Realstatement and proof · cited by 25,697
NormedAddCommGroupstatement and proof · cited by 15,752
NormedSpacestatement and proof · cited by 12,499
Complexstatement and proof · cited by 5,565
nhdsproof · cited by 5,554
Norm.normproof · cited by 5,413
mul_oneproof · cited by 3,885
Filter.Tendstoproof · cited by 3,814
one_mulproof · cited by 2,841
zero_addproof · cited by 2,366
Dist.diststatement and proof · cited by 1,539

Cited by3

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Complex.norm_fderiv_le_div_of_mapsTo_ballproof · cited by 2
Complex.dist_le_dist_of_mapsTo_ballproof · cited by 2
Complex.norm_dslope_le_div_of_mapsTo_ballproof · cited by 1