A cyclic mixed reciprocal sum with a pair-product correction

Proved

WorkbookSource.plus_6808

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

lean-workbooksource-checked

The following inequality a bit of stronger. Let $a$ , $b$ and $c$ be positive numbers such that $a+b+c=3.$ Prove that:

\frac{a}{b+c^2}+\frac{b}{c+a^2}+\frac{c}{a+b^2}+\frac{2(ab+bc+ca)}{9}\geq\frac{13}{6}

Source: InternLM Lean-Workbook, record lean_workbook_plus_6808 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_6808 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a / (b + c^2) + b / (c + a^2) + c / (a + b^2) + (2 * (a * b + b * c + c * a)) / 9 ≥ 13 / 6   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_6808; Apache-2.0

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