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First of all, AM is always greater than or equal to GM $$x_1 + x_2 + ... + x_n \geq \sqrt[n]{x_1x_2...x_n}~\forall x_i \geq 0$$ You can prove it by induction from $\frac{x_1 + x_2}{2} \geq \sqrt{x_1x_2}$ or put $f(x) = \ln(x), p_i = \frac{1}{n}$ to Jensen's inequality to get it. The equality holds when $x_1 = x_2 = ... = x_n$. For author 1 and 2, We ...