I have arrived to this equation in several contexts within branching processes. It arises from textbook exercises, so it must be solvable somehow. Here $f$ is a probability generating function which is known, and $g$ is one which I want to know. Here is a specific case:
$1-g\left(b+\frac{(1-b)(1-a)s}{1-as}\right)=\left(\frac{1-b}{1-a}\right)(1-g(s))$
Where $a$ and $b$ are fixed real numbers.
In this case $g$ should be the pgf of a geometric distribution.
I would write my ideas but I have none. I wouldn't know how to start solving this, and I'm pretty sure there's a small set of solutions for $g$. Thanks in advance for your hints or solutions!