A shifted quadratic ratio upper bound at fixed sum three

Proved

WorkbookSource.base_28549

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

lean-workbooksource-checked

Prove $\frac{a^2}{2a+1}+\frac{b^2}{2b+1}+\frac{c^2}{2c+1}\leq\frac{3(a^2+b^2+c^2)}{2(a^2+b^2+c^2)+3}$ where $a, b, c > 0$ and $a+b+c=3$ .

Source: InternLM Lean-Workbook, record lean_workbook_28549 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_28549 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : a^2 / (2 * a + 1) + b^2 / (2 * b + 1) + c^2 / (2 * c + 1) ≤ 3 * (a^2 + b^2 + c^2) / (2 * (a^2 + b^2 + c^2) + 3)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_28549; Apache-2.0

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