I need to find conditins under which the following holds: Suppose you have two sequences $x_i$ and $y_i$, with $i=1,\ldots, n$, $y_i\ge 0,\;\forall i$ and $\sum_{i=1}^{n}x_i\ge 0$, then what are the minimal (less stringent conditions) for $\sum_{i=1}^{n}x_i y_i\ge 0$?
This problem is an important step in proving that an equilibrium variable increases with a parameter, using the Implicit function theorem and Cramer's rule. That is, I have a system $Az=b$, where $A$ is a $B$-matrix, $z$ is the equilibrium variable and $b$ is a column vector.
$\sum_{i=1}^{n}x_i\ge 0$ follows from the fact that is the column sum of the cofactor metrics ($x_i$ being the cofactors for a given column), which in my case is a B-matrix. B-matrices have positive determinants, positive principal minors, and a positive column sum of cofactors.
By summation by parts I could find a lower bound on $\sum_{i=1}^{n}x_i$ that ensures positiveness of $\sum_{i=1}^{n}x_i y_i\ge 0$. If I sort $y_i$ in ascending order, this lower bound is given by $$-\frac{y_n-y_o}{y_0}\min_{k}\sum_{k=i+1}^{n}x_i.$$ I don't think that this is the least lower bound, but I have not been able to improve on this, nor I have been able to impose conditions in vectors $x$ and $y$ to prove my result.