Is anyone aware of any general (or perhaps not so general) relationship (inequality for instance) relating
$A(x,y)= \frac{\sum_z f(x,y,z)}{\sum_z g(y,z)}$
and
$B(x,y)= \sum_z\left(\frac{f(x,y,z)}{g(y,z)}\right)$
?
Specific context (for what I'm dealing with, but not necessarily the question) is that $\sum_{x,y,x}f(x,y,z)=1$ and $\sum_{y,x}g(y,z)=1$ and $f(x,y,z)\geq 0$ and $g(y,z)\geq 0\quad \forall x,y,z$. I.e. probabilities (or more generally, I guess, measures).
It seems like it could 'vaguely' be related to log sum inequalities (when transformed) or Jensen's inequality perhaps?
One example would be Jensen's inequality:
https://en.wikipedia.org/wiki/Jensen%27s_inequality