A mixed quadratic ratio sum bounded by pairwise ratios

Proved

WorkbookSource.base_39288

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

lean-workbooksource-checked

If $a,b,c>0$ , then

$\frac{a^2+bc}{2a+b+c} + \frac{b^2+ca}{a+2b+c} + \frac{c^2+ab}{a+b+2c}\leq \frac{1}{2}(\frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} + \frac{a+b+c}{2})$

Source: InternLM Lean-Workbook, record lean_workbook_39288 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_39288 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b * c) / (2 * a + b + c) + (b^2 + c * a) / (a + 2 * b + c) + (c^2 + a * b) / (a + b + 2 * c) ≤ (1 / 2) * (a^2 / (b + c) + b^2 / (c + a) + c^2 / (a + b) + (a + b + c) / 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_39288; Apache-2.0

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