I'm dealing with a probability problem and I have to understand the following operation on polynomials: let $F$ and $G$ be any two polynomials of variable $p\in [0,1]$ (to be thought of as a Bernoulli parameter).
What can you say about the "interpolated" polynomial $p\cdot F(p)+(1-p)\cdot G(p)$? Is there a standard word denoting this operation?
How can I find references about that? The problem is that when googling "interpolation of polynomials" I mainly find results on polynomial interpolation!
In my case, the two polynomials induce orientation-preserving homeomorphisms of the closed interval $[0,1]$. Is the resulting interpolation verifying this hypothesis?
I can also suppose that $F$ and $G$ both have a unique fixed point other than $0$ and $1$. What about the fixed points of their interpolation?