A cyclic quartic upper bound by a squared pair-product expression

Proved

WorkbookSource.base_17596

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

lean-workbooksource-checked

Let $a,$ $b,$ $c$ be nonnegative real numbers such that $a+b+c=3$ and $ab+bc+ca>0.$ Prove that $2\left(a^3b+b^3c+c^3a\right)+3abc\le(6-ab-bc-ca)^2.$

Source: InternLM Lean-Workbook, record lean_workbook_17596 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_17596 (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (habc : a + b + c = 3) (h : a * b + b * c + c * a > 0) : 2 * (a ^ 3 * b + b ^ 3 * c + c ^ 3 * a) + 3 * a * b * c ≤ (6 - 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_17596; Apache-2.0

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