A cyclic quartic expression bounds a fourth power of the sum

Proved

WorkbookSource.base_7194

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

lean-workbooksource-checked

Prove that $x(x+y)^3+y(y+z)^3+z(z+x)^3 \ge \frac{8}{27} (x+y+z)^4$ for real numbers $x, y, z$ .

Source: InternLM Lean-Workbook, record lean_workbook_7194 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_7194 (x y z : ℝ) : x * (x + y) ^ 3 + y * (y + z) ^ 3 + z * (z + x) ^ 3 ≥ (8 / 27) * (x + y + z) ^ 4  :=  by sorry

Source