  We apply general methods to generate upper and lower bounds for the essential
minimum of a specific family of height functions. In particular, the results
shown in this article apply to the case of the Zhang-Zagier height.
Furthermore, we can find intervals, where the images of these heights are
dense. Our main tool to find upper bounds and intervals of density, is a
refinement of the classical Fekete-Szego theorem due to Burgos Gil, Philippon,
Rivera-Letelier and Sombra.
