A cyclic ratio bound with a squared difference correction

Proved

WorkbookSource.base_15960

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

lean-workbooksource-checked

Let $a,b,c>0$ . Prove that: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{3}{2}+\frac{\left (a-b \right )^{2}}{2\left (a+b \right )^{2}}$

Source: InternLM Lean-Workbook, record lean_workbook_15960 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_15960 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (b + c) + b / (c + a) + c / (a + b)) ≥ 3 / 2 + (a - b) ^ 2 / (2 * (a + b) ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_15960; Apache-2.0

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