Theorems · Theorem · complex analysis
Complex.norm_dslope_le_div_of_mapsTo_ball
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {R₁ R₂ : ℝ} {f : ℂ → E} {c z : ℂ},
DifferentiableOn ℂ f (Metric.ball c R₁) →
Set.MapsTo f (Metric.ball c R₁) (Metric.closedBall (f c) R₂) → z ∈ Metric.ball c R₁ → ‖dslope f c z‖ ≤ R₂ / R₁Two cases of the Schwarz Lemma (derivative and distance), merged together.
If f : ℂ → E is a complex analytic function on an open ball ball c R₁
hat sends it to a closed ball closedBall (f c) R₂, then the norm of dslope f c z,
which is defined as (z - c)⁻¹ • (f z - f c) for z ≠ c and as deriv f c for z = c,
is not greater than the ratio R₂ / R₁.
- Defined in
- Mathlib.Analysis.Complex.Schwarz
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 296 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites24
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- 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
- Norm.normstatement and proof · cited by 5,413
- Dist.distproof · cited by 1,539
- eq_or_neproof · cited by 1,117
- Metric.ballstatement and proof · cited by 735
- Set.MapsTostatement and proof · cited by 732
- Metric.closedBallstatement and proof · cited by 704
- DifferentiableOnstatement and proof · cited by 419
Cited by1
Results whose statement or proof uses this declaration.
- Complex.affine_of_mapsTo_ball_of_norm_dslope_eq_divproof · cited by 2