A cyclic squared product ratio bounds half the cubic sum

Proved

WorkbookSource.base_57004

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

lean-workbooksource-checked

Prove that, for every positive real numbers $a,b,c$ , the following inequality always holds: $\frac{a^2b^2}{a+b}+\frac{b^2c^2}{b+c}+\frac{c^2a^2}{c+a}\le\frac{a^3+b^3+c^3}2.$

Source: InternLM Lean-Workbook, record lean_workbook_57004 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_57004 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 * b^2 / (a + b) + b^2 * c^2 / (b + c) + c^2 * a^2 / (c + a)) ≤ (a^3 + b^3 + c^3) / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_57004; Apache-2.0

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