Solve the functional equation $$f(ax)=\frac{1}{e^{a+b-x}+e^{bx}}h(x)+f(bx)$$ for $f(x)$ where $h(x)$ is a given function?
You can use any conditions on $f$ and $h$ (like continuity, differentiablity etc) as far as I get some 'nice' non-trivial solution.
Perhaps one of the ways is to exploit the Taylor series and compare the coefficients which is not an efficient way.