A fourth-power ratio with a triple-product correction

Proved

WorkbookSource.base_7522

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

lean-workbooksource-checked

Let a,b,c >0. Prove that: $\frac{ a^4+b^4+c^4}{ab+bc+ca}+\frac{3abc}{a+b+c}\geq \frac{2}{3}(a^2+b^2+c^2)$ ( by Schur )

Source: InternLM Lean-Workbook, record lean_workbook_7522 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_7522 (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) + (3 * a * b * c) / (a + b + c) ≥ (2 / 3) * (a^2 + b^2 + c^2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_7522; Apache-2.0

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