A mixed quadratic reciprocal sum has a symmetric upper bound

Proved

WorkbookSource.plus_39505

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

lean-workbooksource-checked

For $a, b, c>0$ , prove that $\frac{1}{a^2+2bc}+\frac{1}{b^2+2ca}+\frac{1}{c^2+2ab}\leq\frac{ab+bc+ca}{a^2bc+ab^2c+abc^2}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_39505 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_39505 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 / (a ^ 2 + 2 * b * c) + 1 / (b ^ 2 + 2 * c * a) + 1 / (c ^ 2 + 2 * a * b)) ≤ (a * b + b * c + c * a) / (a ^ 2 * b * c + a * b ^ 2 * c + a * b * c ^ 2)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_39505; Apache-2.0

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