A cyclic quartic difference ratio bounds half the pair-product sum

Proved

WorkbookSource.plus_54894

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

lean-workbooksource-checked

For positive real numbers, show that:

$\frac{a^3(b+c-a)}{a^2+bc}+\frac{b^3(c+a-b)}{b^2+ca}+\frac{c^3(a+b-c)}{c^2+ab}\le \frac{ab+bc+ca}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_54894 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_54894 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^3 * (b + c - a) / (a^2 + b * c) + b^3 * (c + a - b) / (b^2 + c * a) + c^3 * (a + b - c) / (c^2 + a * b)) ≤ (a * b + b * c + c * a) / 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_54894; Apache-2.0

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