A refined squared cyclic ratio inequality

Proved

WorkbookSource.base_36251

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

lean-workbooksource-checked

Suppose $a,b,c\in \mathbb R^+$ . Prove that : $\left(\frac ab+\frac bc+\frac ca\right)^2\geq (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)+ \frac{(a-b)^2}{bc+ca+ab} .$

Source: InternLM Lean-Workbook, record lean_workbook_36251 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_36251 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / b + b / c + c / a) ^ 2 ≥ (a + b + c) * (1 / a + 1 / b + 1 / c) + (a - b) ^ 2 / (b * c + c * a + a * b)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_36251; Apache-2.0

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