On a particular extension of the EV-Theorem

The main aim of the paper is to determine the extreme values of the product $P=a_1a_2\cdots a_n$ under the constraints $\sum_{i=1}^n a_i=S$ and $\sum_{i=1}^{n}\frac 1{a_i+1}=S_0$ for $n\ge 3$ nonnegative real numbers $a_1,a_2,\ldots, a_n$ and some given constants $S$ and $S_0$. Some interesting applications of our results are provided as well.