A cyclic ratio sum dominates a pairwise reciprocal sum

Proved

WorkbookSource.base_215

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

lean-workbooksource-checked

The following inequality is stronger, but also easy. Let $a$ , $b$ and $c$ be positive numbers. Prove that:

\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq\frac{3}{2}+\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}

Source: InternLM Lean-Workbook, record lean_workbook_215 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_215 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b + b / c + c / a ≥ 3 / 2 + a / (b + c) + b / (a + c) + c / (a + b)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_215; Apache-2.0

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