A cyclic quadratic ratio upper bound involving symmetric sums

Proved

WorkbookSource.base_16702

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

lean-workbooksource-checked

Given $a,b,c>0$ . Prove that $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\le \frac{3(a^3+b^3+c^3)}{2(ab+bc+ca)}$

Source: InternLM Lean-Workbook, record lean_workbook_16702 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_16702 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 / (b + c) + b^2 / (c + a) + c^2 / (a + b)) ≤ (3 * (a^3 + b^3 + c^3)) / (2 * (a * b + b * c + a * c))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_16702; Apache-2.0

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