A sixth-power bound for three quadratic factors

Proved

WorkbookSource.base_5407

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

lean-workbooksource-checked

For $x,y,z>0$ ,prove that:

$27(x+y+z)^6\geq (27x^2+(y-z)^2)(27y^2+(z-x)^2)(27z^2+(x-y)^2)$

Source: InternLM Lean-Workbook, record lean_workbook_5407 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_5407 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : 27 * (x + y + z) ^ 6 ≥ (27 * x ^ 2 + (y - z) ^ 2) * (27 * y ^ 2 + (z - x) ^ 2) * (27 * z ^ 2 + (x - y) ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_5407; Apache-2.0

View graph