A comparison of shifted linear reciprocal sums

Proved

WorkbookSource.base_15249

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

lean-workbooksource-checked

Let $a,b,c$ be positive real numbers. Prove that $\frac{1}{4a+b+c}+\frac{1}{4b+c+a}+\frac{1}{4c+a+b}+\frac{3}{a+b+c}\le\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}$ .

Source: InternLM Lean-Workbook, record lean_workbook_15249 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_15249 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 / (4 * a + b + c) + 1 / (4 * b + c + a) + 1 / (4 * c + a + b) + 3 / (a + b + c)) ≤ (1 / (a + b) + 1 / (b + c) + 1 / (c + a))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_15249; Apache-2.0

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