A refined comparison of symmetric pairwise ratio sums

Proved

WorkbookSource.base_9865

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

lean-workbooksource-checked

Let $a$ , $b$ and $c$ be positive numbers. Prove that:

\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c} \ge 4\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right)+\frac{2(a^2+b^2+c^2-ab-ac-bc)}{(a+b+c)^2}

Source: InternLM Lean-Workbook, record lean_workbook_9865 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_9865 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (b + c) / a + (c + a) / b + (a + b) / c ≥ 4 * (a / (b + c) + b / (c + a) + c / (a + b)) + (2 * (a ^ 2 + b ^ 2 + c ^ 2 - a * b - a * c - b * c)) / (a + b + c) ^ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_9865; Apache-2.0

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