Minimizing the product of some variables with constant summation having an additional condition

Question

What is the minimum of $a_1\times a_2 \times \dots \times a_n$ such that $a_1+a_2+\dots+a_n=S$ and $0 < x \le a_i \le (1+\alpha)\frac{S}{n}$?

My conjecture is that we need to set as many $a_i$'s as possible to $(1+\alpha)\frac{S}{n}$ and set the rest of $a_i$'s equally. Is that correct? How to prove it?

Answer 1

Set $a_1=0$. $ {} $ $ {} $ $ {} $ $ {} $