A mixed quadratic reciprocal upper bound

Proved

WorkbookSource.base_11443

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

lean-workbooksource-checked

Let $x, y, z$ be positive real numbers. Prove that $\sum_{cyc}{\frac{1}{xy+2z^2}} \leq \frac{xy+yz+zx}{xyz(x+y+z)}.$

Source: InternLM Lean-Workbook, record lean_workbook_11443 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_11443 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : 1 / (x * y + 2 * z ^ 2) + 1 / (y * z + 2 * x ^ 2) + 1 / (z * x + 2 * y ^ 2) ≤ (x * y + y * z + z * x) / (x * y * z * (x + y + z))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_11443; Apache-2.0

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