A cyclic product-power ratio comparison at fixed sum three

Proved

WorkbookSource.base_2775

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

lean-workbooksource-checked

Let $x,y,z$ be positive reals with sum $3$ . Prove: $\displaystyle{\frac{y^3z^3+2}{yz}+\frac{z^3x^3+2}{zx}+\frac{x^3y^3+2}{xy}\ge \frac{x^2+2}{x}+\frac{y^2+2}{y}+\frac{z^2+2}{z}}$

Source: InternLM Lean-Workbook, record lean_workbook_2775 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_2775 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 3) : (y^3 * z^3 + 2) / (y * z) + (z^3 * x^3 + 2) / (z * x) + (x^3 * y^3 + 2) / (x * y) ≥ (x^2 + 2) / x + (y^2 + 2) / y + (z^2 + 2) / z  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_2775; Apache-2.0

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