A cyclic cubic ratio upper bound with a quadratic correction

Proved

WorkbookSource.base_17694

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

lean-workbooksource-checked

Let $a,b,c>0$ . Prove $3(a+b+c)\ge \frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}+\frac{8(ab+bc+ca)}{a+b+c}$

Source: InternLM Lean-Workbook, record lean_workbook_17694 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_17694 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : 3 * (a + b + c) ≥ a^3 / (a^2 + a * b + b^2) + b^3 / (b^2 + b * c + c^2) + c^3 / (c^2 + c * a + a^2) + 8 * (a * b + b * c + c * a) / (a + b + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_17694; Apache-2.0

View graph