  Two hypotheses on the class $\mathcal{L}(\gamma)$ in the class
$\mathcal{OS}\cap \mathcal{ID}$ are discussed. Two weak hypotheses on the class
$\mathcal{L}(\gamma)$ in the class $\mathcal{OS}\cap \mathcal{ID}$ are proved.
A necessary and sufficient condition in order that, for every $t >0$, the
$t$-th convolution power of a distribution in the class $\mathcal{OS}\cap
\mathcal{ID}$ belongs to the class $\mathcal{L}(\gamma)$ is given.
