  Let B=A+K where A is a bounded selfadjoint operator and K is an element of
the von Neumann-Schatten ideal S_p with p>1. Let {\lambda_n} denote an
enumeration of the discrete spectrum of B. We show that $\sum_n
\dist(\lambda_n, \sigma(A))^p$ is bounded from above by a constant multiple of
|K|_p^p. We also derive a unitary analog of this estimate and apply it to
obtain new estimates on zero-sets of Cauchy transforms.
