A squared symmetric cubic expression is bounded by a cubic square sum

Proved

WorkbookSource.plus_38199

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

lean-workbooksource-checked

Let a,b,c>0, prove

$\left(abc + (a + b)(b + c)(c + a) \right)^2 \le 3(a^2 + b^2 + c^2)^3$

Source: InternLM Lean-Workbook, record lean_workbook_plus_38199 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_38199 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b * c + (a + b) * (b + c) * (c + a))^2 ≤ 3 * (a^2 + b^2 + c^2)^3   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_38199; Apache-2.0

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