A quartic bound for the squared pairwise sum

Proved

WorkbookSource.base_16069

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Prove that for positive numbers $a, b, c$ , the following inequality holds: $(a^3+b^3+c^3+9abc)(a+b+c) \ge 4 (ab+bc+ca)^2$ .

Source: InternLM Lean-Workbook, record lean_workbook_16069 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_16069 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^3 + b^3 + c^3 + 9 * a * b * c) * (a + b + c) ≥ 4 * (a * b + b * c + c * a)^2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_16069; Apache-2.0

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