A quartic sum and triple product bound a cubic sum

Proved

WorkbookSource.plus_30461

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

lean-workbooksource-checked

Let $a,b,c$ be real (not only positive) numbers such that $a+b+c = 1$ . Prove the inequality

$2(a^4+b^4+c^4) + abc \ge a^3 + b^3 + c^3$

Source: InternLM Lean-Workbook, record lean_workbook_plus_30461 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_30461 (a b c : ℝ) (ha : a + b + c = 1) : 2 * (a^4 + b^4 + c^4) + a * b * c ≥ a^3 + b^3 + c^3   :=  by sorry

Source