lean_workbook_plus_12705

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \ge 4(x + y + z) \implies xy + yz+ zx \ge 2\left( {x + y + z} \right) \implies \sqrt x + \sqrt y + \sqrt z \le \frac{3}{2}\sqrt {xyz}$

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_12705 (x y z : ℝ) (hx : x > 0 ∧ y > 0 ∧ z > 0) (hab : x + y + z = 1) : 1 / x + 1 / y + 1 / z ≥ 4 * (x + y + z) → (x * y + y * z + z * x ≥ 2 * (x + y + z)) → (Real.sqrt x + Real.sqrt y + Real.sqrt z ≤ 3 / 2 * Real.sqrt (x * y * z))   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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