On the distribution of $\phi (\sigma (n))$

Let $\phi (n)$ be the Euler totient function and $\sigma (n)$ denote the sum of divisors of $n$. In this note, we obtain explicit upper bounds on the number of positive integers $n\le x$ such that $\phi (\sigma (n))>cn$ for any $c>0$. This is a refinement of a result of Alaoglu and Erdős.