le_rpow_one_add_norm_iff_norm_le — Mathlib

English

Let a,c ∈ ℂ and R ∈ ℝ with R ≠ 0. Then circleAverage(log ‖· − a‖) c R = log R + log⁺ (R⁻¹ ‖c − a‖).

Русский

Пусть a,c ∈ ℂ и R ∈ ℝ, R ≠ 0. Тогда circleAverage(log ‖· − a‖) c R = log R + log⁺ (R⁻¹ ‖c − a‖).

LaTeX

$$$$R \neq 0 \implies circleAverage(\log \|\cdot - a\|)\, c\, R = \log R + \log^+ (R^{-1} \|c - a\|).$$$$

Lean4

theorem le_rpow_one_add_norm_iff_norm_le {r t : ℝ} (hr : 0 < r) (ht : 0 < t) (x : E) :
    t ≤ (1 + ‖x‖) ^ (-r) ↔ ‖x‖ ≤ t ^ (-r⁻¹) - 1 :=
  by
  rw [le_sub_iff_add_le', neg_inv]
  exact (Real.le_rpow_inv_iff_of_neg (by positivity) ht (neg_lt_zero.mpr hr)).symm

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