A cyclic quadratic-difference ratio lower bound

Proved

WorkbookSource.base_9286

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

The following inequality is also true. Let $a, b, c>0$ . Prove that $\sum_{cyc}{\frac{a^2-bc}{b^2+c^2}}\ge 3\left(1-\frac{ab+bc+ca}{a^2+b^2+c^2}\right)$

Source: InternLM Lean-Workbook, record lean_workbook_9286 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_9286 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 - b * c) / (b^2 + c^2) + (b^2 - c * a) / (c^2 + a^2) + (c^2 - a * b) / (a^2 + b^2) ≥ 3 * (1 - (a * b + b * c + c * a) / (a^2 + b^2 + c^2))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_9286; Apache-2.0

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