A parameterized weighted reciprocal sum comparison

Proved

WorkbookSource.base_16519

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

lean-workbooksource-checked

Let $x,y,z,k > 0$ , prove that $\frac{1}{kx + y + z} + \frac{1}{x + ky + z} + \frac{1}{x + y + kz} \leq \frac{1}{k + 2} \cdot \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)$

Source: InternLM Lean-Workbook, record lean_workbook_16519 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_16519 (x y z k : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (hk : 0 < k) : 1 / (k * x + y + z) + 1 / (x + k * y + z) + 1 / (x + y + k * z) ≤ 1 / (k + 2) * (1 / x + 1 / y + 1 / z)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_16519; Apache-2.0

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