A quadratic reciprocal sum bounds a ratio of quadratic and cubic sums

Proved

WorkbookSource.plus_82476

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

lean-workbooksource-checked

Prove that $\frac{a}{a^2+bc}+\frac{b}{b^2+ca}+\frac{c}{c^2+ab} \geq \frac{3(a^2+b^2+c^2)}{2(a^3+b^3+c^3)}$

Hold for positive real $a,b,c$

Source: InternLM Lean-Workbook, record lean_workbook_plus_82476 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_82476 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (a ^ 2 + b * c) + b / (b ^ 2 + c * a) + c / (c ^ 2 + a * b)) ≥ 3 * (a ^ 2 + b ^ 2 + c ^ 2) / (2 * (a ^ 3 + b ^ 3 + c ^ 3))   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_82476; Apache-2.0

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