This paper is concerned with
the behavior of certain combinations of generalized means of positive real numbers,
considered as functions of the index set. It is shown that these combinations are
actually superadditive functions (over set unions) of the index set. Several previously
established inequalities of this nature are obtained as corollaries of the main theorem,
namely, certain results of R. Rado, W. N. Everitt, D. S. Mitrinovič and P. M.
Vasič, and H. Kestleman.