Zeros of Meromorphic Functions with Poles Close to the Real Axis

Abstract.

Let

$$ f(z) = \sum\limits_{k = 1}^\infty {\frac{{a_k }} {{z - z_k }}} ,\quad\mathop {\lim }\limits_{k \to \infty } z_k = \infty ,\quad\sum\limits_{z_k \ne 0} {\left| {\frac{{a_k }} {{z_k }}} \right|} < \infty $$

. If the z _k lie sufficiently close to the real axis and the a _k are close in an average sense to the positive real axis then f has infinitely many zeros.