A cyclic pairwise ratio sum has a normalized quadratic lower bound

Proved

WorkbookSource.plus_78986

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

lean-workbooksource-checked

Let $a,b,c>0$ . Prove $\frac{3(a^2+b^2+c^2)}{4(a+b+c)^2}+\frac{11}{4}\le \frac{a+b}{b+c}+\frac{b+c}{c+a}+\frac{c+a}{a+b}$

Source: InternLM Lean-Workbook, record lean_workbook_plus_78986 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_78986 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (3 * (a ^ 2 + b ^ 2 + c ^ 2) / (4 * (a + b + c) ^ 2) + 11 / 4) ≤ (a + b) / (b + c) + (b + c) / (c + a) + (c + a) / (a + b)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_78986; Apache-2.0

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