A squared cyclic ratio sum bounded by a normalized eighth-degree expression

Proved

WorkbookSource.base_40891

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

lean-workbooksource-checked

Let $a, b, c>0$ . Prove that $\left(a+\frac{b^2}{c}\right)^2+\left(b+\frac{c^2}{a}\right)^2+\left(c+\frac{a^2}{b}\right)^2\le\frac{4(a^2+b^2+c^2)^4}{27a^2b^2c^2}$

Source: InternLM Lean-Workbook, record lean_workbook_40891 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_40891 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b^2 / c)^2 + (b + c^2 / a)^2 + (c + a^2 / b)^2 ≤ (4 * (a^2 + b^2 + c^2)^4) / (27 * a^2 * b^2 * c^2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_40891; Apache-2.0

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