A refined quadratic reciprocal product inequality

Proved

WorkbookSource.plus_14551

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

lean-workbooksource-checked

For $a,b,c>0$ .Prove that $\left( {{a^2} + {b^2} + {c^2}} \right)\left( {\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} + \frac{1}{{{c^2}}}} \right) + 5 \ge \frac{{14({a^2} + {b^2} + {c^2})}}{{ab + bc + ca}}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_14551 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_14551 (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) + 5 ≥ 14 * (a^2 + b^2 + c^2) / (a * b + b * c + c * a)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_14551; Apache-2.0

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