Zeros of linear partial fractions I am interested in somehow characterizing the zeros of the following function:
$P(z) = \sum_{i=1}^n \frac{\alpha_i}{z+\lambda_i}$, 
with $\lambda_1\geq \lambda_2 \geq \ldots \geq \lambda_n=0$ and $\sum_{i=1}^n \alpha_i = 0$.
Furthermore, the values $\alpha_i$ are in general non-integer and may even be complex.  The poles $\lambda_i$, however, are all real.
I initially had hoped to use something like the Gauss-Lucas theorem for this, but the conditions on $\alpha_i$ seem to prevent this.
While it would be great to make some strong statements about the zeros, I would be more then satisfied to find conditions guaranteeing that the zeros lie in the left-half of the complex plane.
Any thoughts/suggestions would be great.  Thanks!
 A: For given $\lambda_i$, you have $n$ non-zero, linearly independent functions and you form a linear combination of them with arbitrary complex coefficients $\alpha_i$ -- that means you can pick $n-1$ points in general position and make them zeros of $P$ by a suitable choice of the coefficients (e.g. by picking one of the functions and mimicking its function values at the $n-1$ chosen points by a suitable linear combintaion of the other $n-1$ functions). Conversely, since you can bring all the fractions to a common denominator and the numerator will then be of degree $n-1$, the function has exactly $n-1$ zeros in the general case. So generally you have $n-1$ zeros that can be almost anywhere you like (except for the poles, and sets of measure zero where the function values are linearly dependent), and the only thing that's left to characterize is what sort of special cases you can get.
(I implicitly assumed that the $\lambda_i$ are different and the $\alpha_i$ are non-zero -- if either of these assumptions isn't fulfilled, you can reduce to a case with lower $n$ where they are.)
