enorm_fderiv_norm_rpow_le — Mathlib

English

In the ENorm setting, the derivative bound for norm^p holds with ENNReal scaling and conversions between ENorm and norm.

Русский

В ENorm установке борняы для norm^p работают с масштабированием ENNReal и переходами между ENorm и нормой.

LaTeX

$$enorm_fderiv_norm_rpow hf hp$$

Lean4

theorem enorm_fderiv_norm_rpow_le {f : F → E} (hf : Differentiable ℝ f) {x : F} {p : ℝ≥0} (hp : 1 < p) :
    ‖fderiv ℝ (fun x ↦ ‖f x‖ ^ (p : ℝ)) x‖ₑ ≤ p * ‖f x‖ₑ ^ ((p : ℝ) - 1) * ‖fderiv ℝ f x‖ₑ := by
  simpa [enorm, ← ENNReal.coe_rpow_of_nonneg _ (sub_nonneg.2 <| NNReal.one_le_coe.2 hp.le), ← ENNReal.coe_mul] using
    nnnorm_fderiv_norm_rpow_le hf hp

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