On the ratio between two factorization functions

Let $f(n)$ be the number of factorizations of $n$, i.e., representations of $n$ as an unordered product of integers greater than $1$. In addition, let $F(n)$ be the number of factorizations of $n$ into distinct parts. In this note, we provide a new upper bound for the ratio $f(n)/F(n)$.