A product of weighted quadratic forms bounds symmetric sums

Proved

WorkbookSource.base_47159

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

lean-workbooksource-checked

Given $a, b, c > 0$ . Prove that: $(a^2 + 2b^2)(b^2 + 2c^2)(c^2 + 2a^2) \geq\ \frac {1}{3}(ab + bc + ca)^2(a + b + c)^2$

Source: InternLM Lean-Workbook, record lean_workbook_47159 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_47159 {a b c : ℝ} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + 2 * b^2) * (b^2 + 2 * c^2) * (c^2 + 2 * a^2) ≥ 1/3 * (a * b + b * c + c * a)^2 * (a + b + c)^2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_47159; Apache-2.0

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