A mixed quartic ratio sum bounds the pair-product sum

Proved

WorkbookSource.plus_68899

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

lean-workbooksource-checked

For all positive real numbers $a,b,c$ we have $\frac{a^{4}+b^{2}c^{2}}{c^{2}+a^{2}}+\frac{b^{4}+c^{2}a^{2}}{a^{2}+b^{2}}+\frac{c^{4}+a^{2}b^{2}}{b^{2}+c^{2}} \geq ab+bc+ca$

Source: InternLM Lean-Workbook, record lean_workbook_plus_68899 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_68899 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^4 + b^2 * c^2) / (c^2 + a^2) + (b^4 + c^2 * a^2) / (a^2 + b^2) + (c^4 + a^2 * b^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_plus_68899; Apache-2.0

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