A pairwise ratio sum bounded by cyclic ratios

Proved

WorkbookSource.base_35902

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

lean-workbooksource-checked

Prove(using only AM-GM if possible) that for a,b,c>0 $\sum_{cyc}\frac{c}{a+b}\leq\frac{3}{2}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-2\right)$

Source: InternLM Lean-Workbook, record lean_workbook_35902 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_35902 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (c / (a + b) + b / (a + c) + a / (b + c)) ≤ (3 / 2) * (a / b + b / c + c / a - 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_35902; Apache-2.0

View graph