A shifted cyclic ratio upper bound at fixed sum three

Proved

WorkbookSource.base_8025

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

lean-workbooksource-checked

For $a, b, c>0, a+b+c=3$ prove that $\frac{a}{a+b+1}+\frac{b}{b+c+1}+\frac{c}{c+a+1}\le\frac{(a^2+b^2+c^2)^2}{9}$

Source: InternLM Lean-Workbook, record lean_workbook_8025 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_8025 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a / (a + b + 1) + b / (b + c + 1) + c / (c + a + 1) ≤ (a ^ 2 + b ^ 2 + c ^ 2) ^ 2 / 9  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_8025; Apache-2.0

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