A pairwise ratio sum with a fourth-degree correction

Proved

WorkbookSource.base_16854

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

lean-workbooksource-checked

Let $a$ , $b$ and $c$ positive real numbers. Prove that: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \geq \frac{3}{2}+\frac{4(a^4+b^4+c^4-abc(a+b+c))}{(a+b+c)^4}$

Source: InternLM Lean-Workbook, record lean_workbook_16854 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_16854 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + b / (c + a) + c / (a + b)) ≥ 3 / 2 + 4 * (a ^ 4 + b ^ 4 + c ^ 4 - a * b * c * (a + b + c)) / (a + b + c) ^ 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_16854; Apache-2.0

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