A pairwise product ratio sum has a symmetric quadratic upper bound

Proved

WorkbookSource.plus_23206

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{ab(a+b)}{2ab+ac+bc}+\frac{ac(a+c)}{2ac+ab+bc}+\frac{bc(b+c)}{2bc+ac+ab}\leq\frac{3(a^2+b^2+c^2+ab+ac+bc)}{4(a+b+c)}

Source: InternLM Lean-Workbook, record lean_workbook_plus_23206 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_23206 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b * (a + b) / (2 * a * b + a * c + b * c) + a * c * (a + c) / (2 * a * c + a * b + b * c) + b * c * (b + c) / (2 * b * c + a * c + a * b)) ≤ (3 * (a ^ 2 + b ^ 2 + c ^ 2 + a * b + a * c + b * c)) / (4 * (a + b + c))   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_23206; Apache-2.0

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