A squared reciprocal sum inequality at fixed sum three

Proved

WorkbookSource.plus_65265

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

lean-workbooksource-checked

For $a,b,c,>0$ such that $a+b+c=3$ Prove that

3\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-1\right)^2+1\ge \frac{4}{abc}+3\left(\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}\right)

Source: InternLM Lean-Workbook, record lean_workbook_plus_65265 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_65265 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : 3 * (1 / a + 1 / b + 1 / c - 1) ^ 2 + 1 ≥ 4 / (a * b * c) + 3 * (a / (b * c) + b / (c * a) + c / (a * b))   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_65265; Apache-2.0

View graph