A quadratic product ratio has a symmetric upper bound

Proved

WorkbookSource.plus_67957

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

lean-workbooksource-checked

Prove that for positive numbers $a, b, c$ , $\frac{a^2b^2c^2}{(a^2+bc)(b^2+ca)(c^2+ab)}\le\frac{3(ab+bc+ca)(a+b+c)}{8(a+b+c)^3}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_67957 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_67957 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b * c) ^ 2 / ((a ^ 2 + b * c) * (b ^ 2 + a * c) * (c ^ 2 + a * b)) ≤ (3 * (a * b + b * c + a * c) * (a + b + c)) / (8 * (a + b + c) ^ 3)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_67957; Apache-2.0

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