A cubic correction to a quadratic norm

Proved

WorkbookSource.base_10651

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

lean-workbooksource-checked

Let $a, b, c > 0$ so that $a + b + c = 1$ . Prove that $3(a^2 + b^2 + c^2) - 2(a^3 + b^3 + c^3) \geq \frac{7}{9}$ .

Source: InternLM Lean-Workbook, record lean_workbook_10651 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_10651 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 1) : 3 * (a ^ 2 + b ^ 2 + c ^ 2) - 2 * (a ^ 3 + b ^ 3 + c ^ 3) ≥ 7 / 9  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_10651; Apache-2.0

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