I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form
$$R(z):= \sum\_{j=1}^{n} \frac{a\_j}{z-p\_j},$$
where the $a\_j$'s are nonzero complex numbers and the $p\_j$'s are distinct complex numbers.


Any relevant reference is welcome.


Note : The [Gauss-Lucas theorem](http://en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem) implies that if the $a\_j$'s are all assumed to be positive, then the roots of $R$ lie in the convex hull of the set of poles $\{p\_1,\dots, p\_n\}$. However, this is false in the general case, but perhaps there is something to be said.


Thank you,
Malik



On the geometry of roots of a sum of complex linear fractions
Section 21 in Chapter V of "Geometry of Polynomials" by Morris Marden, AMS, Mathematical Surveys 3, 1966 may contain what you are looking for, Lemma 21.1.on page 98 is as follows:


Let $C\_j(z) := |z - c\_j|^2 - r\_j^2$ and $C\_j$ be the circular region defined by $\pm C\_j(z) \leq 0$, let $p\_j$ be in $C\_j$ and let $a\_j$ be real or complex constants, then every root $Z$ of 


$$ \sum\_{j=1}^n \frac{a\_j}{z - p\_j} = 0 $$


is either contained in one of the $C\_j$ or satisfies the inequality


$$ \left| \sum\_{j=1}^n \frac{a\_j(c\_j - Z)}{C\_j(Z)} \right|^2 - \left( \sum\_{j=1}^n \frac{|a\_j|r\_j}{|C\_j(Z)|} \right)^2 \ \leq \ 0 \, .$$


While it is not explicitly stated there, I suppose the $c\_j$ are arbitrary complex numbers, while the $r\_j$ are positive real numbers. I adapted the notation to your question and fixed what I considered a minor typo, but did no thorough proofreading. 


