A cyclic cubic-over-linear sum bounds the quadratic mean

Proved

WorkbookSource.base_11617

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

lean-workbooksource-checked

For positive real numbers $x, y, z$ , prove the inequality:

\displaylines {\frac {x ^ 3} {x + y} + \frac {y ^ 3} {y + z} + \frac {z ^ 3} {z + x} \geq \frac {x^2+y^2+z^2} {2}.}

Source: InternLM Lean-Workbook, record lean_workbook_11617 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_11617 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (x ^ 3 / (x + y) + y ^ 3 / (y + z) + z ^ 3 / (z + x)) ≥ (x ^ 2 + y ^ 2 + z ^ 2) / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_11617; Apache-2.0

View graph