A cyclic linear ratio sum bounded by the normalized quadratic sum

Proved

WorkbookSource.base_15933

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

lean-workbooksource-checked

Let $a,b,c>0$ . Prove that:

$\frac{a+2b}{2a+b}+\frac{b+2c}{2b+c}+\frac{c+2a}{2c+a}\le \frac{9(a^2+b^2+c^2)}{(a+b+c)^2}$

Source: InternLM Lean-Workbook, record lean_workbook_15933 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_15933 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + 2 * b) / (2 * a + b) + (b + 2 * c) / (2 * b + c) + (c + 2 * a) / (2 * c + a) ≤ (9 * (a ^ 2 + b ^ 2 + c ^ 2)) / (a + b + c) ^ 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_15933; Apache-2.0

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