A product of sums bounds two symmetric quadratic ratios

Proved

WorkbookSource.base_19044

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

lean-workbooksource-checked

If $a,b,c$ are positive real numbers, then $(a+b+c)\left ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right ) \geqslant{\frac{27}{2}\frac{a^2+b^2+c^2}{(a+b+c)^2}+\frac{3}{2}\frac{(a+b+c)^2}{ab+bc+ca}}$

Source: InternLM Lean-Workbook, record lean_workbook_19044 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_19044 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a + b + c) * (1 / a + 1 / b + 1 / c) ≥ (27 / 2 * (a ^ 2 + b ^ 2 + c ^ 2) / (a + b + c) ^ 2 + 3 / 2 * (a + b + c) ^ 2 / (a * b + b * c + a * c))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_19044; Apache-2.0

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