A cyclic product ratio bounded by symmetric quadratic forms

Proved

WorkbookSource.base_30484

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

lean-workbooksource-checked

Let $a,b,c$ be possitive real numbers,prove that,

$\frac{a+b}{b+c}\frac{a}{2a+b+c}+\frac{b+c}{c+a}\frac{b}{2b+c+a}+\frac{c+a}{a+b}\frac{c}{2c+a+b}\le \frac{3}{4}\frac{a^2+b^2+c^2}{ab+bc+ca}$

Source: InternLM Lean-Workbook, record lean_workbook_30484 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_30484 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b) / (b + c) * (a / (2 * a + b + c)) + (b + c) / (c + a) * (b / (2 * b + c + a)) + (c + a) / (a + b) * (c / (2 * c + a + b)) ≤ 3 / 4 * (a ^ 2 + b ^ 2 + c ^ 2) / (a * b + b * c + c * a)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_30484; Apache-2.0

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