SFMT

Theorem

Given nonnegative real numbers $e_1, e_2, e_3, \cdots ,e_n$ and that $x_1+x_2+x_3+\cdots+x_n$ is fixed and all the terms inside the sum are nonnegative, the maximum value of $x_1^{e_1}x_2^{e_2}x_3^{e_3}\cdots x_n^{e_n}$ is when $\dfrac{x_1}{e_1}=\dfrac{x_2}{e_2}=\dfrac{x_3}{e_3}=\cdots =\dfrac{x_n}{e_n}.$

Proof

The weighted AM-GM Inequality states that if $a_1, a_2, \dotsc, a_n$ are nonnegative real numbers, and $\lambda_1, \lambda_2, \dotsc, \lambda_n$ are nonnegative real numbers (the "weights") which sum to 1, then $\lambda_1 a_1 + \lambda_2 a_2 + \dotsb + \lambda_n a_n \ge a_1^{\lambda_1} a_2^{\lambda_2} \dotsm a_n^{\lambda_n}.$ We let $\lambda_i=\dfrac{e_i}{e_1+e_2+\cdots+e_n}$ and $a_i=\dfrac{x_i}{e_i}$ in this inequality. We get that $x_1+x_2+x_3+\cdots+x_n \ge \sqrt[n]{\dfrac{x_1^{e_1}x_2^{e_2}x_3^{e_3}\cdots x_n^{e_n}}{e_1^{e_1}e_2^{e_2}e_3^{e_3}\cdots e_n^{e_n}}}(e_1+e_2+e_3+\cdots+e_n).$ Dividing both sides and then taking to the nth power, we get $\left(\dfrac{x_1+x_2+x_3+\cdots+x_n}{e_1+e_2+e_3+\cdots+e_n}\right)^n \ge \dfrac{x_1^{e_1}x_2^{e_2}x_3^{e_3}\cdots x_n^{e_n}}{e_1^{e_1}e_2^{e_2}e_3^{e_3}\cdots e_n^{e_n}}.$ Then we can multiply both sides to get $x_1^{e_1}x_2^{e_2}x_3^{e_3}\cdots x_n^{e_n} \le \left(\dfrac{x_1+x_2+x_3+\cdots+x_n}{e_1+e_2+e_3+\cdots+e_n}\right)^n (e_1^{e_1}e_2^{e_2}e_3^{e_3}\cdots e_n^{e_n}).$ The equality case for the weighted AM-GM inequality is when all the $a_i$ terms such that $e_i$ is not $0$ are equal, or in this case $\dfrac{x_1}{e_1}=\dfrac{x_2}{e_2}=\dfrac{x_3}{e_3}=\cdots=\dfrac{x_n}{e_n}.$