A symmetric quadratic-product ratio sum bounds the quadratic mean

Proved

WorkbookSource.base_2714

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

lean-workbooksource-checked

Given $a, b, c > 0$ , prove that

$\frac{ab(a^2+b^2)}{a^2+b^2+ac+bc}+\frac{bc(b^2+c^2)}{b^2+c^2+ba+ca}+\frac{ca(c^2+a^2)}{c^2+a^2+cb+ab}\le \frac{a^2+b^2+c^2}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_2714 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_2714 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :  (a * b * (a ^ 2 + b ^ 2) / (a ^ 2 + b ^ 2 + a * c + b * c) + b * c * (b ^ 2 + c ^ 2) / (b ^ 2 + c ^ 2 + b * a + c * a) + c * a * (c ^ 2 + a ^ 2) / (c ^ 2 + a ^ 2 + c * b + a * b)) ≤ (a ^ 2 + b ^ 2 + c ^ 2) / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_2714; Apache-2.0

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