  Let $X$ and $Y$ be two independent random variables with corresponding
distributions $F$ and $G$ supported on $[0,\infty)$. The distribution of the
product $XY$, which is called the product convolution of $F$ and $G$, is
denoted by $H$. In this paper, some suitable conditions about $F $ and $G $ are
given, under which the distribution $H$ belongs to the long-tailed distribution
class. Here, $F$ is a generalized long-tailed distribution and is not
necessarily an exponential distribution. Finally, a series of examples are
given to show that the above conditions are satisfied by many distributions and
one of them is necessary in some sense.
