A squared shifted ratio upper bound at fixed sum three

Proved

WorkbookSource.base_53649

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

lean-workbooksource-checked

Let $a, b, c > 0$ such that $a+b+c=3$ . Prove that $\dfrac{a^{2}}{(2a+1)^2} + \dfrac{b^{2}}{(2b+1)^2} + \dfrac{c^{2}}{(2c+1)^2} \leq \dfrac{a^{2}+b^{2}+c^{2}}{a^{2}+b^{2}+c^{2}+6}$

Source: InternLM Lean-Workbook, record lean_workbook_53649 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_53649 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a + b + c = 3) : (a^2 / (2 * a + 1)^2 + b^2 / (2 * b + 1)^2 + c^2 / (2 * c + 1)^2) ≤ (a^2 + b^2 + c^2) / (a^2 + b^2 + c^2 + 6)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_53649; Apache-2.0

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