A product of sums bounds a symmetric product ratio

Proved

WorkbookSource.base_28443

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

lean-workbooksource-checked

If $x, y, z$ are positive real numbers then: $(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \ge 7+ \frac{2(xy+yz+zx)^2}{3xyz(x+y+z)}$

Source: InternLM Lean-Workbook, record lean_workbook_28443 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_28443 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x + y + z) * (1 / x + 1 / y + 1 / z) ≥ 7 + 2 * (x * y + y * z + z * x) ^ 2 / (3 * x * y * z * (x + y + z))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_28443; Apache-2.0

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