Question

Functional equations of the form $f(g(x))=(f(x))^p$, where $g(x)$ is known, is called Bottcher equation. Generally, we have only asymptotic formula for the solution $f(x)$ under certain conditions. In one of my study, I come across the following problem,

How to find a nontrivial (i.e., nonconstant) function $f(x)\in C^\infty(0 ,+\infty)$ satisfying the following functional equation $$f(\frac{2x^3+a}{3x^2})=f(x)^2,$$ where $a>0$ is a constant.

Or can you tell me finding a explicit form for $f(x)$ is impossible?

Answer 1

Introduce the new unknown function $$g(t):=\log f(a^{1/3}(1+t)).$$ It satisfies a functional equation of the type $$g(t)=g(t^2(1+o(1))/2$$ for $t$ near 0, and $g(0)=0$. I guess from this you should be able to prove that $g(t)$ is identically $0$.

Answer 2

In my answer $x \mapsto f(x)$ is the function satisfying your functional equation and $g: t \mapsto g(t)$ is defined explicitly in terms of $f$. After some computation you will see that $g$ satisfies a functional equation of the stated type. Begin with $$\exp(2 g(t))=(f(a^{1/3}(1+t)))^2= \ldots$$