An arithmetic-harmonic mean bound with a squared-difference correction

Proved

WorkbookSource.base_1603

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

lean-workbooksource-checked

Let $a,b,c>0$ . Prove that

\\frac{a+b+c}{3}\\ge \\frac{3}{\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}}+\\frac{(a-b)^2}{3(a+b+c)}

Source: InternLM Lean-Workbook, record lean_workbook_1603 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_1603 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b + c) / 3 ≥ 3 / (1 / a + 1 / b + 1 / c) + (a - b) ^ 2 / (3 * (a + b + c))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1603; Apache-2.0

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