A weighted quadratic reciprocal upper bound

Proved

WorkbookSource.base_15307

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

lean-workbooksource-checked

for positive real numbers $x,y,z$ prove that

\frac{x}{4x^2+4y^2+z^2}+\frac{y}{4y^2+4z^2+x^2}+\frac{z}{4z^2+4x^2+y^2} \leq \frac{1}{x+y+z}

Source: InternLM Lean-Workbook, record lean_workbook_15307 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_15307 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x / (4 * x ^ 2 + 4 * y ^ 2 + z ^ 2) + y / (4 * y ^ 2 + 4 * z ^ 2 + x ^ 2) + z / (4 * z ^ 2 + 4 * x ^ 2 + y ^ 2)) ≤ 1 / (x + y + z)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_15307; Apache-2.0

View graph