Given $a>0$, $b>0$, I am trying to find the function $f_{a,b} : \mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that for all $u \in \mathbb{R}_+$, $$\exp\left\{\;\int\limits_{\mathbb{R}_+} \ln\left(1+uy\right) f_{a,b}(y) \partial y\right\} = \int\limits_{\mathbb{R}_+} \left(1+ux\right)^{-a} \frac{x^{b-1}e^{-x}}{\Gamma(b)} \partial x.$$ The right side integral is the Laplace transform of the product of two gamma distributions with shapes $a$, $b$ and scales 1 (you can recognise $\frac{x^{b-1}e^{-x}}{\Gamma(b)}$ as the density of a gamma). On the left, you have the general form of a GGC random variable, where $f_{a,b}$ is the density of some measure over $\mathbb{R}_+$ (which is not needed to integrate to 1, it can even integrate to $\infty$).
According to Bondesson - A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables, we have existence and unicity of the function $f_{a,b}$, for all $a$, $b$.
$f_{a,b}$ is not needed to be continuous, although I suspect it'll be.
The question is the following: Is there some known theory about these kind of problems? How is this kind of "integral condition over a function" even called? How should I proceed to find a solution?
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