A quadratic ratio inequality with a normalized product correction

Proved

WorkbookSource.plus_74161

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

lean-workbooksource-checked

If $a,\,b,\,c$ are positive real numbers, then (1). $\frac{a^2+b^2+c^2}{ab+bc+ca} + 1 \geqslant \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}+\frac{4abc}{(a+b)(b+c)(c+a)}.$

Source: InternLM Lean-Workbook, record lean_workbook_plus_74161 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_74161 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^2 + b^2 + c^2) / (a * b + b * c + a * c) + 1 ≥ a / (a + b) + b / (b + c) + c / (c + a) + 4 * a * b * c / ((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_74161; Apache-2.0

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