A cyclic sixth-degree pair-product ratio bounds a quartic sum

Proved

WorkbookSource.base_28714

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

lean-workbooksource-checked

For $a, b, c, d>0$ prove that ${\frac{a^4c^2+b^4d^2}{cd}+\frac{b^4d^2+c^4a^2}{da}+\frac{c^4a^2+d^4b^2}{ab}+\frac{d^4b^2+a^4c^2}{bc}\geq4(a^2c^2+b^2d^2)}$

Source: InternLM Lean-Workbook, record lean_workbook_28714 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_28714 (a b c d : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) : (a^4 * c^2 + b^4 * d^2) / (c * d) + (b^4 * d^2 + c^4 * a^2) / (d * a) + (c^4 * a^2 + d^4 * b^2) / (a * b) + (d^4 * b^2 + a^4 * c^2) / (b * c) ≥ 4 * (a^2 * c^2 + b^2 * d^2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_28714; Apache-2.0

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