A symmetric shifted reciprocal sum bounded by a cubic ratio

Proved

WorkbookSource.base_2207

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

lean-workbooksource-checked

The following inequality a bit of stronger. Let $a$ , $b$ and $c$ be positive numbers. Prove that:

\frac{1}{7a+b+c} +\frac{1}{a+7b+c} +\frac{1}{a+b+7c} \leq \frac{3(a^2+b^2+c^2)}{(a+b+c) ^3}.

Source: InternLM Lean-Workbook, record lean_workbook_2207 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_2207 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 / (7 * a + b + c) + 1 / (a + 7 * b + c) + 1 / (a + b + 7 * c)) ≤ (3 * (a ^ 2 + b ^ 2 + c ^ 2) / (a + b + c) ^ 3)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_2207; Apache-2.0

View graph