A cyclic cubic ratio with a squared pair-product correction

Proved

WorkbookSource.plus_28468

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

lean-workbooksource-checked

2/Let $a,\, b, \, c>0$ . Prove

{\frac {{a}^{3}}{b}}+{\frac {{b}^{3}}{c}}+{\frac {{c}^{3}}{a}}+{\frac { \left( ab+ca+bc \right) ^{2}}{{a}^{2}+{b}^{2}+{c}^{2}}}\ge \frac{2 \left( a+b+c \right) ^{2}}{3}

Source: InternLM Lean-Workbook, record lean_workbook_plus_28468 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_28468 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^3 / b + b^3 / c + c^3 / a + (a * b + b * c + c * a)^2 / (a^2 + b^2 + c^2)) ≥ (2 * (a + b + c)^2) / 3   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_28468; Apache-2.0

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