Consider the following function $$ f(x) = x\cdot h(x) - g(h(x)) $$ where $x\in\mathbb{R}, f:\mathbb{R} \to \mathbb{R}^{+}$ and $f, h, g$ are analytic everywhere and extend to be complex analytic on $\mathbb{C}$
I'm interested in the imaginary component of the continuation of $f$, $f:\mathbb{C} \to \mathbb{C}$, that is
$$ f(z) = u(x,y) + i\;v(x,y), \quad z = x + iy $$ where I want to be able to write out $v$ in terms of $g$ and $h$, or at least be able to give some properties of $v$.
I'm not sure why I'm finding this so difficult, presumably because I require extra conditions on $h(x)$ and $g(x)$? For example, I think (this is somewhere upstream in the overall argument I'm trying to make) I can safely say that $h(x)$ is strictly increasing.
My question is therefore - assuming I had extra conditions on the function $g(x)$, how would I go about finding $v(x,y)$ without having an explicit form for $h(x)$ and $g(x)$, and instead just knowing general properties?