A symmetric quartic bound with coefficient twenty-seven

Proved

WorkbookSource.plus_56211

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

lean-workbooksource-checked

Let $a,b,c \geq 0$ ,prove that: $8(b^2+c^2+a^2)^2+27(a+b+c)abc \geq 9(ab+cb+ac)(b^2+c^2+a^2)+3(a+b+c)(a+b)(b+c)(c+a)$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_56211 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_56211 (a b c : ℝ) : 8 * (b ^ 2 + c ^ 2 + a ^ 2) ^ 2 + 27 * (a + b + c) * a * b * c ≥ 9 * (a * b + b * c + c * a) * (b ^ 2 + c ^ 2 + a ^ 2) + 3 * (a + b + c) * (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_56211; Apache-2.0

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