A mixed quadratic reciprocal sum bounds pairwise reciprocals

Proved

WorkbookSource.base_5378

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

lean-workbooksource-checked

prove that : $\frac{1}{x^2+yz}+\frac{1}{y^2+xz}+\frac{1}{z^2+xy}\leq\,\frac{1}{2}(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{xz})$ where $x,y,z\succ0$

Source: InternLM Lean-Workbook, record lean_workbook_5378 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_5378 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (1 / (x ^ 2 + y * z) + 1 / (y ^ 2 + z * x) + 1 / (z ^ 2 + x * y)) ≤ (1 / 2) * (1 / (x * y) + 1 / (y * z) + 1 / (x * z))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_5378; Apache-2.0

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