could you please help me on the following problem?
I have the characteristic functions of two random variables $X$ and $Y$, denoted $\phi_X(u)$ and $\phi_Y(u)$, such that: \begin{equation*} \phi_X(u) = \left(a + (1-a)\phi_Y(u) \right)^b \end{equation*} with $0<a<1$ and $b>0$. I know the density of $Y$ and would like somehow to know the one of $X$.
If $1/2\le a<1$, I can write: \begin{equation*} \phi_X(u) = a\left(1 + \frac{(1-a)}{a}\phi_Y(u) \right)^b \end{equation*} and I can apply the generalized binomial theorem because $\frac{(1-a)}{a}<1$ and $|\phi_Y(u)|<\le 1$. Because I know the law of $Y$ I can get a sort of infinite mixture of densities that is fine for me.
However, I do not know how to proceed in case $0<a<1/2$.
Do you have some hints or ideas?
Thanks a lot in advance for your help.