A cyclic quartic ratio sum is bounded by the quadratic sum

Proved

WorkbookSource.plus_80477

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

lean-workbooksource-checked

Prove that for all positive real numbers $\sum \frac{a^3(b+c)}{a^2+bc}+\frac{b^3(c+a)}{b^2+ac}+\frac{c^3(a+b)}{c^2+ab}\le a^2+b^2+c^2$

Source: InternLM Lean-Workbook, record lean_workbook_plus_80477 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_80477 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^3 * (b + c) / (a^2 + b * c) + b^3 * (c + a) / (b^2 + a * c) + c^3 * (a + b) / (c^2 + a * b)) ≤ 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_80477; Apache-2.0

View graph