Given $n$-vectors $x, y_1, y_2$ such that the Spearman correlation coefficient of $x$ and $y_i$ is $\rho_i = \rho(x,y_i)$, are there known bounds on the Spearman coefficient of $x$ with $y_1 + y_2$, in terms of the $\rho_i$ (and $n$, presumably)? That is, can one find (non-trivial) functions $l(\rho_1,\rho_2,n), u(\rho_1,\rho_2,n)$ such that $$l(\rho_1,\rho_2,n) \le \rho(x,y_1+y_2) \le u(\rho_1,\rho_2,n)$$
edit: per @whuber's example in the comment, it appears that in the general case, only the trivial bounds $l = -1, u = 1$ can be made. Thus, I would like to further impose the constraint:
- $y_1, y_2$ are permutations of the integers $1 \ldots n$.