of the above proof.

From this, we obtain an analogue result to Corollary \[cor:densitySD\] characterizing the densities of distributions in ${\mathrm{T}}$.

\[cor:densityGGC\] Let $f(t)$ be a probability density with support $[a, \infty)$, $a\geq 0$, which is continuous on $(a, \infty)$. Then $f$ is the density of a generalized gamma convolution, if and only if $f$ fulfills $$\begin{aligned}
 (a-t) f(t) + \int_a^t f(s) \int_{t-s}^\infty m(x) dx\, ds=0, \quad t\geq a,\end{aligned}$$ for some $m(x):(0,\infty)\to [0,\infty)$ which is CM and such that $\int_0^\infty \log^+(x) m(x) dx<\infty$.

The statement follows similarly to Corollary \[cor:densitySD\] with the help of Proposition \[prop:bondessontoGGC\].

As mentioned, the $c$-factors of selfdecomposable distributions play an important role for our studies. In the following proposition, which is of interest by its own, we will see, that the GGCs are exactly those distributions in ${\mathrm{L}}^+$ whose $c$-factors are all in Bondesson’s class. Its proof is postponed to the closing section of this article.

\[prop:GGCfactorinBO\] Let $\mu\in{\mathrm{T}}$, then $\mu_c\in{\mathrm{BO}}$ for all $c>0$, $c\neq 1$. Conversely, if $\mu\in{\mathrm{L}}^+$ with either $\mu_c\in{\mathrm{BO}}$ for all $c\in(0,1)$, or $\mu_c\in{\mathrm{BO}}$ for all $c>1$, then $\mu\in{\mathrm{T}}$.

Summarizing, we can state the characterizations of the class ${\mathrm{T}}$ similarly to that of ${\mathrm{L}}^+$ in Proposition \[prop:SD\].

\[cor:GGC\] Let $\mu\in {\mathrm{L}}^+$ be 