A normalized product correction bounds a quadratic ratio

Proved

WorkbookSource.plus_1587

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

lean-workbooksource-checked

If $a, b, c>0$ prove that $\frac{27abc}{(a+b+c)^3}+2\geq\frac{3(ab+bc+ca)}{a^2+b^2+c^2}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_1587 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_1587 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (27 * a * b * c) / (a + b + c) ^ 3 + 2 ≥ (3 * (a * b + b * c + a * c)) / (a ^ 2 + b ^ 2 + c ^ 2)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_1587; Apache-2.0

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