A weighted cyclic cubic ratio bounds a quadratic expression

Proved

WorkbookSource.plus_55238

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

lean-workbooksource-checked

Prove that for positive numbers $a, b, c$ , the following inequality holds: $4\left(\frac{(3a+b)a^2}{b+c} + \frac{(3b+c)b^2}{c+a} + \frac{(3c+a)c^2}{a+b}\right) \geq 11(a^2+b^2+c^2) - 3(ab+bc+ca)$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_55238 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_55238 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 4 * ((3 * a + b) * a ^ 2 / (b + c) + (3 * b + c) * b ^ 2 / (c + a) + (3 * c + a) * c ^ 2 / (a + b)) ≥ 11 * (a ^ 2 + b ^ 2 + c ^ 2) - 3 * (a * b + b * c + c * a)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_55238; Apache-2.0

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