A fourth-power bound for a squared pairwise sum

Proved

WorkbookSource.base_15608

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

lean-workbooksource-checked

For x,y,z positive reals such that x+y+z=1 prove that $x^4+y^4+z^4 +\frac 1{27} \geq \frac 23 (xy+yz+zx)^2 .$

Source: InternLM Lean-Workbook, record lean_workbook_15608 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_15608 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 1) :  x ^ 4 + y ^ 4 + z ^ 4 + 1 / 27 ≥ 2 / 3 * (x * y + y * z + z * x) ^ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_15608; Apache-2.0

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