A shifted cyclic ratio has a symmetric quadratic upper bound

Proved

WorkbookSource.plus_3792

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

lean-workbooksource-checked

If $a,b,c$ are postive reals such that $a+b+c=3$ , prove that

\frac{a}{1+ab}+\frac{b}{1+bc}+\frac{c}{1+ca}\leq \frac{3}{2}\frac{a^2+b^2+c^2}{ab+bc+ca}

Source: InternLM Lean-Workbook, record lean_workbook_plus_3792 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_3792 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a + b + c = 3) : (a / (1 + a * b) + b / (1 + b * c) + c / (1 + c * a)) ≤ (3 / 2) * (a ^ 2 + b ^ 2 + c ^ 2) / (a * b + b * c + c * a)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_3792; Apache-2.0

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