A weighted cubic ratio bounds a quadratic difference

Proved

WorkbookSource.base_25430

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

lean-workbooksource-checked

The following inequality is also true. Let $a$ , $b$ and $c$ be positive numbers. Prove that: $\frac{a^{2}b}{4a+7b}+\frac{b^{2}c}{4b+7c}+\frac{c^{2}a}{4c+7a}\ge\frac{5(ab+bc+ca)-2(a^2+b^2+c^2)}{33}$

Source: InternLM Lean-Workbook, record lean_workbook_25430 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_25430 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 * b / (4 * a + 7 * b) + b^2 * c / (4 * b + 7 * c) + c^2 * a / (4 * c + 7 * a)) ≥ (5 * (a * b + b * c + c * a) - 2 * (a^2 + b^2 + c^2)) / 33  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_25430; Apache-2.0

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