A squared cyclic ratio sum bounds a product of sums

Proved

WorkbookSource.base_26054

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

lean-workbooksource-checked

If $x,y,z>0$ then: $5(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})^2 \geq 2(x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})+27$

Source: InternLM Lean-Workbook, record lean_workbook_26054 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_26054 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : 5 * (x / y + y / z + z / x) ^ 2 ≥ 2 * (x + y + z) * (1 / x + 1 / y + 1 / z) + 27  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_26054; Apache-2.0

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