A weighted quadratic ratio upper bound under a product constraint

Proved

WorkbookSource.base_23382

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

lean-workbooksource-checked

The following inequality is also true. If $a, b, c>0$ and $abc=1$ prove that $\frac{7ab+bc+ca}{a^2+b^2+c^2+3bc}+\frac{ab+7bc+ca}{a^2+b^2+c^2+3ca}+\frac{ab+bc+7ca}{a^2+b^2+c^2+3ab}\leq\frac{9}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_23382 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_23382 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a * b * c = 1) : (7 * a * b + b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2 + 3 * b * c) + (a * b + 7 * b * c + c * a) / (a ^ 2 + b ^ 2 + c ^ 2 + 3 * c * a) + (a * b + b * c + 7 * c * a) / (a ^ 2 + b ^ 2 + c ^ 2 + 3 * a * b) ≤ 9 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_23382; Apache-2.0

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