A pair-product squared ratio upper bound

Proved

WorkbookSource.base_28080

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

lean-workbooksource-checked

Let $a, b, c>0$ . Prove that $\frac{(a+b)(b+c)}{(2a+b+c)^2}+\frac{(b+c)(c+a)}{(2b+c+a)^2}+\frac{(c+a)(a+b)}{(2c+a+b)^2}\le\frac{(a+b+c)^2}{4(ab+bc+ca)}$

Source: InternLM Lean-Workbook, record lean_workbook_28080 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_28080 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) * (b + c) / (2 * a + b + c) ^ 2 + (b + c) * (c + a) / (2 * b + c + a) ^ 2 + (c + a) * (a + b) / (2 * c + a + b) ^ 2 ≤ (a + b + c) ^ 2 / (4 * (a * b + b * c + c * a))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_28080; Apache-2.0

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