A squared quadratic sum times reciprocals bounds a cubic expression

Proved

WorkbookSource.base_4117

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

lean-workbooksource-checked

For any positive real numbers $a,b$ and $c$ , $(a^{2} + b^{2} + c^{2})^{2}\left(\frac {1}{a} + \frac {1}{b} + \frac {1}{c}\right) + 27abc\ge 2(a + b + c)^{3}(*)$

Source: InternLM Lean-Workbook, record lean_workbook_4117 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_4117 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b^2 + c^2)^2 * (1 / a + 1 / b + 1 / c) + 27 * a * b * c ≥ 2 * (a + b + c)^3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_4117; Apache-2.0

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