After some work, I got this nice inequality:
$$ \frac{n^2}{2} < \phi(n)\cdot \sigma(n) $$
where $\phi(n)$ is Euler's phi function and $\sigma(n)= \sum_{d|n} d$. I know this is true because I'm aware that this can be further refined to
$$ \frac{6 n^2}{\pi^2} < \phi(n)\cdot \sigma(n) $$
However, I'm interested in the first one because I'm sure there is an elemental proof of it (which I can't find at the moment). Any ideas?
If $n=\prod_ip_i^{a_i}$, then $$ \sigma(n)=\prod_i \frac{p_i^{a_i+1}-1}{p_i-1}=n\prod_i\frac{1-p_i^{-a_i-1}}{1-p_i^{-1}}, $$ and $$ \phi(n)=n\prod_i(1-p_i^{-1}) $$ Hence we obtain $$ \frac{\sigma(n)\phi(n)}{n^2}=\prod_i (1-p_i^{-a_i-1}). $$ Hence the first inequality is obvious, and the second also: each of the exponents is less than or equal to $−2$, so the product is at least as large as the product $\prod_p(1 − p^{−2})=\frac{1}{\zeta(2)}=\frac{6}{\pi^2}$. Hence we obtain $$ 6\frac{n^2}{\pi^2}<\sigma(n)\phi(n). $$ The first inequality is obtained if we just use $\frac{1}{\zeta(2)}>\frac{1}{2}$.
