Theorems · Theorem · complex analysis
Complex.dist_le_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] {R : ℝ} {f : E → F} {c z : E},
DifferentiableOn ℂ f (Metric.ball c R) →
Set.MapsTo f (Metric.ball c R) (Metric.closedBall (f c) R) → z ∈ Metric.ball c R → dist (f z) (f c) ≤ dist z cThe Schwarz Lemma. Let f : E → F be a complex analytic function
on an open ball with center c.
If f sends this ball to a closed ball with center f c and the same radius,
then for any z in the former ball we have dist (f z) (f c) ≤ dist z c.
- Defined in
- Mathlib.Analysis.Complex.Schwarz
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 294 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
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
- one_mulproof · cited by 2,841
- Dist.diststatement and proof · cited by 1,539
- LT.lt.ne'proof · cited by 1,417
- 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 by2
Results whose statement or proof uses this declaration.
- Complex.norm_le_norm_of_mapsTo_ballproof · cited by 1
- Complex.dist_le_dist_of_mapsTo_ball_selfproof · cited by 0