Fix $N \geq 2$, $k \in [0,1]$, and define $$F_k(x_1,\ldots,x_N) = \sum (1+x_i) - \frac{(N-k)^2}{\sum (1+x_i)^{-1}}.$$ In general, $x_i > y_i$ for all $i$ does not imply $F_k(x_1,\ldots,x_N) > F_k(y_1,\ldots,y_N)$ (take $(.91,.1,.81)$ and $(.9,0,.8)$ and $k = .5$, for example). I'm interested in characterizing the region of $[0,1]^N$ over which $F_k$ is increasing.
I can see that $F_k(x_1,\ldots,x_N) > F_k(y_1,\ldots,y_N)$ if and only if $$\frac{\sum (1+x_i) - \sum(1+y_i)}{(N-k)^2} > \frac{1}{\sum (1+x_i)^{-1}} - \frac{1}{\sum (1+y_i)^{-1}}.$$
But I'm wondering if there is anything more illuminating to be said here about conditions under which the difference between $\sum (1+x_i)$ and $\sum (1+y_i)$ exceeds that between $(\sum (1+x_i)^{-1})^{-1}$ and $(\sum (1+y_i)^{-1})^{-1}$ by a factor of $(N-k)^2$. More generally, is there anything to be said about how the difference between two sums compares to the difference between the reciprocals of the sums of the reciprocals?