In this Wikipedia is defined the so-called Dickman–de Bruijn function. If we assume that $$I:=\int_0^\infty \rho(u)f(u)du$$ is convergent, for example you can think in the function $f(u)=e^{-u^2}$, and we assume that $$J:=\int_0^\infty \rho'(u)f(u)du$$ is also convergent, I've curisosity if there is a way for which one can dilucidate some relationship involving $I$ and $J$.
I don't know if previous exercise was in the literature. In order to establish a clear question, I consider next particular case
Question. Let $\rho(u)$ the Dickman function. If we define $$I:=\int_0^\infty \rho(u)e^{-u^2}du$$ and $$J:=\int_0^\infty \rho'(u)e^{-u^2}du,$$ how can we find a concise relationship between $I$ and $J$? I am asking about identities, or well inequalities involving these real numbers $I$ and $J$. Thanks in advance.
In a concise way, the Dickman function can be defined as the continuous function $\rho:\mathbb{R}^+\to (0,1)$ which equals $1$ on the interval $(0,1)$ and fulfills the differential equation $\rho'(u) = -\frac{1}{u}\rho(u-1)$. In particular $I<\int_{0}^{+\infty}e^{-u^2}\,du=\tfrac{1}{2}\sqrt{\pi}$ is trivial and
$$\begin{eqnarray*} I &=& \int_{0}^{1}e^{-u^2}\,du +\int_{1}^{+\infty}\rho(u)e^{-u^2}\,du\\&=&\sqrt{\pi}\,\text{Erf}(1)+\tfrac{\sqrt{\pi}}{2}\int_{1}^{+\infty}\rho(u-1)\frac{\text{Erf}(u)}{u}\,du\end{eqnarray*} $$ can be deduced from integration parts and leads to better bounds.
$J$ can be managed in a similar way, $$ J=\int_{1}^{+\infty}\rho'(u)e^{-u^2}\,du = -\int_{0}^{+\infty}\rho(u)\frac{e^{-(u+1)^2}}{(u+1)}\,du $$ leads to $J\in\left(-\frac{1}{9},0\right)$ and better bounds can be deduced by applying integration by parts and recalling the differential equation fulfilled by $\rho$.