  Motivated by the large ammount of results obtained for minimal and positive
constant mean curvature surfaces in several ambient spaces, the aim of this
paper is to obtain half-space theorems for properly immersed surfaces in
$\mathbb{R}^3$ whose mean curvature is given as a prescribed function of its
Gauss map. In order to achieve this purpose, we will study the behavior at
infinity of a 1-parameter family of properly embedded annuli that are analogous
to the usual minimal catenoids.
