A symmetric quartic inequality on the unit sphere

Proved

WorkbookSource.base_21546

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

lean-workbooksource-checked

Let $x$ , $y$ and $z$ be real numbers such that $x^{2}+y^{2}+z^{2}=1$ . Prove that: $7(x^{4}+y^{4}+z^{4})+8(x+y+z)(xyz)\ge 4(xy+yz+zx)+1$

Source: InternLM Lean-Workbook, record lean_workbook_21546 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_21546 (x y z : ℝ) (hx : x ^ 2 + y ^ 2 + z ^ 2 = 1) :
  7 * (x ^ 4 + y ^ 4 + z ^ 4) + 8 * (x + y + z) * (x * y * z) ≥
  4 * (x * y + y * z + z * x) + 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_21546; Apache-2.0

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