A squared cyclic ratio sum bounds a symmetric quadratic ratio

Proved

WorkbookSource.plus_9785

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

lean-workbooksource-checked

Let $a,b,c>0.$ Prove that $\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+2\ge \frac{5(a^2+b^2+c^2)}{ab+bc+ca}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_9785 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_9785 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / b^2 + b^2 / c^2 + c^2 / a^2) + 2 ≥ 5 * (a^2 + b^2 + c^2) / (a * b + b * c + a * c)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_9785; Apache-2.0

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