We use weighted Chebyshev's inequality: if $a_1, \dots, a_n > 0$, and $x_1 \le x_2 \le \dots \le x_n$, $y_1 \le y_2 \le \dots \le y_n$, then
$$\frac{(a_1x_1 + \dots + a_nx_n)(a_1y_1 + \dots + a_ny_n)}{a_1 + \dots + a_n} \le a_1x_1y_1 + \dots + a_nx_ny_n. \quad (*)$$

Now let $p(x) = a_1 + a_2x + \dots + a_nx^{n-1}$ be the given polynomial. Then we simply set in $(*)$ $x_1 = 1$, $x_2 = x$, ..., $x_i = x^{i-1}$, $y_1 = 1$, ..., $y_i = y^{i-1}$, thus finishing the proof.