Consider the finite form of Jensen's inequality. For the convex case ($\phi>1$):
$$ \frac{\sum \delta_ix_i^\phi}{\left(\sum \delta_i x_i\right)^\phi} > 1 $$
and for the concave case ($\phi < 1$):
$$ \frac{\sum \delta_ix_i^\phi}{\left(\sum \delta_i x_i\right)^\phi} < 1 $$
where $\sum \delta_i = 1$.
I am interested in the following fraction or relation:
$$ \frac{\sum \delta_i x_i^\phi y_i^\theta}{\left(\sum \delta_i x_i\right)^\phi\left(\sum \delta_i y_i\right)^\theta} $$
In principle there is no constraint in the relationship between $x_i$ and $y_i$, so they could be not independent.
I have a "feeling" the value depends on both $\phi$ and $\theta$, and in the covariance between $x_i$ and $y_i$.
For instance, that fraction, using the Jensen's inequality for the two terms in the denominator, for given values of the exponents, can be related to:
$$ \frac{\sum \delta_i x_i^\phi y_i^\theta}{\sum \delta_ix_i^\phi\sum \delta_iy_i^\theta} $$
It might be reasonable to assume this fraction is close to 1 if variables are uncorrelated (although this might only be true if $\delta_i = 1/n$, i.e. weights are uncorrelated with the two variables).
Maybe this has already been studied, as it "looks" like a straightforward generalisation of Jensen's inequality, but I have not found any other inequality that resembles this. Are you aware of anything providing an analysis of this relationship?