A cyclic cubic-over-quadratic upper bound

Proved

WorkbookSource.base_21170

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

lean-workbooksource-checked

For a,b,c>0. Demonstrate: $\frac{ab^2}{a^2+2b^2+c^2}+\frac{bc^2}{b^2+2c^2+a^2}+\frac{ca^2}{c^2+2a^2+b^2}\le \frac{a+b+c}{4}$

Source: InternLM Lean-Workbook, record lean_workbook_21170 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_21170 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a * b ^ 2 / (a ^ 2 + 2 * b ^ 2 + c ^ 2) + b * c ^ 2 / (b ^ 2 + 2 * c ^ 2 + a ^ 2) + c * a ^ 2 / (c ^ 2 + 2 * a ^ 2 + b ^ 2)) ≤ (a + b + c) / 4  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_21170; Apache-2.0

View graph