A refined comparison of quadratic and linear reciprocal products

Proved

WorkbookSource.base_54617

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

lean-workbooksource-checked

Given $a, b, c$ three positive real numbers, prove that $(a^2+b^2+c^2)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} \right)+27\ge 4(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right)$ without using the UVW method.

Source: InternLM Lean-Workbook, record lean_workbook_54617 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_54617 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b^2 + c^2) * (1 / a^2 + 1 / b^2 + 1 / c^2) + 27 ≥ 4 * (a + b + c) * (1 / a + 1 / b + 1 / c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_54617; Apache-2.0

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