Show that $\phi(n)\sigma(n)<n^2$ ,for all $n>1$.
Continuing from what you've done,
$$\begin{equation}\begin{aligned} \prod_{i=1}^k p^{\alpha_i-1}(p^{\alpha_i+1}-1) & = \prod_{i=1}^k (p^{2\alpha_i}-p^{\alpha_i-1}) \\ & \lt \prod_{i=1}^k p^{2\alpha_i} \\ & = \left(\prod_{i=1}^k p^{\alpha_i}\right)^2 \\ & = n^2 \end{aligned}\end{equation}$$