A sum bound by three shifted squares

Proved

WorkbookSource.plus_4724

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

lean-workbooksource-checked

Let $a = 1 + x^2, b = 1 + y^2, c = 1 + z^2$ for $x, y, z \ge 0$ . The desired inequality is written as

(x + y + z)^2 \le \frac43(1 + x^2)(1 + y^2)(1 + z^2).

Source: InternLM Lean-Workbook, record lean_workbook_plus_4724 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_4724 (x y z : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) : (x + y + z) ^ 2 ≤ (4 / 3) * (1 + x ^ 2) * (1 + y ^ 2) * (1 + z ^ 2)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_4724; Apache-2.0

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