  In this paper, the class of (complex) quasi-Herglotz functions is introduced
as the complex vector space generated by the convex cone of ordinary Herglotz
functions. We prove characterization theorems, in particular, an analytic
characterization. The subclasses of quasi-Herglotz functions that are
identically zero in one half-plane as well as rational quasi-Herglotz functions
are investigated in detail. Moreover, we relate to other areas such as weighted
Hardy spaces, definitizable functions, the Cauchy transform on the unit circle
and sum-rule identities.
