Norm tsum log Deriv expo le of norm le
norm_tsum_logDeriv_expo_le_of_norm_le
Plain-language statement
Monotone version of norm_tsum_logDeriv_expo_le: if ‖q‖ ≤ r < 1, then ‖∑ n·qⁿ/(1-qⁿ)‖ ≤ r/(1-r)³. Useful when we have a uniform bound on ‖q‖.
Exact Lean statement
lemma norm_tsum_logDeriv_expo_le_of_norm_le {q : ℂ} {r : ℝ} (hqr : ‖q‖ ≤ r) (hr : r < 1) :
‖∑' n : ℕ+, (n : ℂ) * q ^ (n : ℕ) / (1 - q ^ (n : ℕ))‖ ≤ r / (1 - r) ^ 3Formal artifact
Lean source
lemma norm_tsum_logDeriv_expo_le_of_norm_le {q : ℂ} {r : ℝ} (hqr : ‖q‖ ≤ r) (hr : r < 1) : ‖∑' n : ℕ+, (n : ℂ) * q ^ (n : ℕ) / (1 - q ^ (n : ℕ))‖ ≤ r / (1 - r) ^ 3 := by have hq : ‖q‖ < 1 := lt_of_le_of_lt hqr hr have hr_nonneg : 0 ≤ r := le_trans (norm_nonneg _) hqr calc ‖∑' n : ℕ+, (n : ℂ) * q ^ (n : ℕ) / (1 - q ^ (n : ℕ))‖ ≤ ‖q‖ / (1 - ‖q‖) ^ 3 := norm_tsum_logDeriv_expo_le hq _ ≤ r / (1 - r) ^ 3 := by have := sub_pos.mpr hr have := sub_pos.mpr hq gcongr- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/Eisenstein.lean:449-458
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Anti Der Pos
antiDerPos
Plain-language statement
If is a modular form where is positive for sufficiently large (i.e. constant term is positive) and the derivative is positive, then is also positive.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Anti Serre Der Pos
antiSerreDerPos
Plain-language statement
Let be a holomorphic function where is real for all . Assume that Serre derivative is positive on the imaginary axis. If is positive for sufficiently large , then is positive for all .
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Closed Ball center subset upper Half Plane
closedBall_center_subset_upperHalfPlane
Plain-language statement
Closed ball centered at z with radius z.im/2 is contained in the upper half plane.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.