This is a follow-up of my previous question which received a lot of very nice and helpful answers. However one question which was actually of equal importance for me was there not answered. Therefore I will ask it again now in a much more broad context.
Let $$\hat f(x,q)=\sum_{k=0}^\infty f(xq^{2k})-f(xq^{2k+1}),\tag1$$where $f$ is a function of a rather general nature which need not be specified here, as my original interest is about analytic functions.
I would like to prove the following conjecture ($a>0$ is assumed):
The function $\hat f(a,q)$ decreases on the whole interval $q\in[0,1)$ if and only if the function $f(x)$ increases on the whole interval $[0,a]$.
This conjecture holds by numerical evidence. Moreover it seems that the number of increasing/decreasing intervals of $\hat f(a,q)$ on the domain $0<q<1$ is in general equal to the number of decreasing/increasing intervals of the original function $f(x)$ on the domain $0<x<a$.
The conjecture receives an additional support from the following observation for analytic functions. In this case $(1)$ is equivalent to:$$\hat f(x,q)=\sum_{k=0}^\infty\frac{f_k x^k}{1+q^k},\tag2$$
From this one obtains:$${\hat f}_q'(x,0)=-xf'(0);\quad \hat f'_q(x,1)=-\frac x4 f'(x),\tag3$$where in the last equality the definition $f'_q(x,1):=\lim_{q\to1^-}f'_q(x,q)$ is assumed. Thus the derivative of $\hat f(x,q)$ with respect to $q$ at the ends of the interval $[0,1)$ has the opposite sign to the derivatives of $f(x)$ with respect to $x$ at the ends of the interval $[0,x]$. However I did not find a way to evaluate the derivative for intermediate values of $q$. So any help is appreciated.