A squared reciprocal sum bounds quadratic reciprocal terms

Proved

WorkbookSource.base_4484

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

lean-workbooksource-checked

Let $a,b,c$ be three positive real numbers , prove that $(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^2\ge\frac{1}{a^2}+\frac{4}{b^2+c^2}+\frac{18}{a^2+b^2+c^2}.$

Source: InternLM Lean-Workbook, record lean_workbook_4484 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_4484 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (1 / a + 1 / b + 1 / c) ^ 2 ≥ 1 / a ^ 2 + 4 / (b ^ 2 + c ^ 2) + 18 / (a ^ 2 + b ^ 2 + c ^ 2)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_4484; Apache-2.0

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