A shifted squared cyclic ratio sum bounds a symmetric ratio

Proved

WorkbookSource.base_27098

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove that $\left(1+\frac{2a}{b}\right)^{2}+\left(1+\frac{2b}{c}\right)^{2}+\left(1+\frac{2c}{a}\right)^{2}\ge\frac{9(a+b+c)^{2}}{ab+bc+ca}.$

Source: InternLM Lean-Workbook, record lean_workbook_27098 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_27098 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 + 2 * a / b) ^ 2 + (1 + 2 * b / c) ^ 2 + (1 + 2 * c / a) ^ 2 ≥ 9 * (a + b + c) ^ 2 / (a * b + b * c + c * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_27098; Apache-2.0

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