Padoa’s inequality is named after Alessandro Padoa (1868-1937):

Let $ a$ , $ b$ , $ c$ be sidelengths of a given triangle $ \triangle ABC$ then

$ $ (b+c-a)(c+a-b)(a+b-c) \le abc .$ $

My question:Is the following generalization of Padoa’s inequality corect?

Let $ a_i>0$ for $ 1\le i\le n$ and let $ $ S:=a_1+a_2+….+a_n.$ $ Suppose that $ $ b_i:=S-(n-1)a_i \ge 0\quad\text{ for} \quad 1\le i\le n.$ $ Then

$ $ \prod_{i=1}^n b_i \leq \prod_{1}^{n} a_i .$ $