A product comparison between fourth-power and quadratic sums

Proved

WorkbookSource.base_37977

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

lean-workbooksource-checked

Let a,b,c>0. Prove that: $\left( {{a^4} + {b^4} + {c^4}} \right)\left( {ab + bc + ca} \right) \ge \left( {{a^2} + {b^2} + {c^2}} \right)\left( {{a^2}{b^2} + {b^2}{c^2} + {c^2}{a^2}} \right)$ . Can you prove it by SOS?

Source: InternLM Lean-Workbook, record lean_workbook_37977 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_37977 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a ^ 4 + b ^ 4 + c ^ 4) * (a * b + b * c + c * a) ≥ (a ^ 2 + b ^ 2 + c ^ 2) * (a ^ 2 * b ^ 2 + b ^ 2 * c ^ 2 + c ^ 2 * a ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_37977; Apache-2.0

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