A pairwise cyclic ratio sum has a symmetric quadratic upper bound

Proved

WorkbookSource.plus_26827

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

lean-workbooksource-checked

The following inequality is also true. Let $a$ , $b$ and $c$ are positive numbers. Prove that: $\frac{2a}{a+b}+\frac{2b}{b+c}+\frac{2c}{c+a} \le \frac{7(a^2+b^2+c^2)+2(ab+ac+bc)}{(a+b+c)^2}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_26827 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_26827 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (2 * a / (a + b) + 2 * b / (b + c) + 2 * c / (c + a)) ≤ (7 * (a ^ 2 + b ^ 2 + c ^ 2) + 2 * (a * b + b * c + c * a)) / (a + b + c) ^ 2   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_26827; Apache-2.0

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