A squared pair sum with a linear correction bounds mixed square-root products

Proved

WorkbookSource.base_21498

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

lean-workbooksource-checked

Prove that for positive numbers a and b, the inequality $\frac{(a+b)^2}{2}+\frac{a+b}{4} \geq a\sqrt{b}+b\sqrt{a}$ holds.

Source: InternLM Lean-Workbook, record lean_workbook_21498 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_21498 (a b : ℝ) (ha : 0 < a) (hb : 0 < b) : (a + b) ^ 2 / 2 + (a + b) / 4 ≥ a * Real.sqrt b + b * Real.sqrt a  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_21498; Apache-2.0

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