Theorems · Theorem · complex analysis
Complex.norm_deriv_le_div_of_mapsTo_ball
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c : ℂ},
DifferentiableOn ℂ f (Metric.ball c R₁) →
Set.MapsTo f (Metric.ball c R₁) (Metric.closedBall (f c) R₂) → 0 < R₁ → ‖deriv f c‖ ≤ R₂ / R₁The Schwarz Lemma: if f : ℂ → E is complex analytic
on an open disk with center c and a positive radius R₁,
and it sends this disk to a closed ball with center f c and radius R₂,
then the norm of the derivative of f at c is at most the ratio R₂ / R₁.
- Defined in
- Mathlib.Analysis.Complex.Schwarz
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 295 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- 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
- Norm.normstatement · cited by 5,413
- Metric.ballstatement and proof · cited by 735
- Set.MapsTostatement and proof · cited by 732
- Metric.closedBallstatement and proof · cited by 704
- derivstatement · cited by 676
- DifferentiableOnstatement and proof · cited by 419
- norm_deriv_eq_norm_fderivproof · cited by 4
- Complex.norm_fderiv_le_div_of_mapsTo_ballproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- Complex.norm_dslope_le_div_of_mapsTo_ballproof · cited by 1
- Complex.norm_deriv_le_one_of_mapsTo_ballproof · cited by 0