On generalized means and generalized convex functions

Properties of generalized convex functions, defined in terms of the generalized means introduced by Hardy, Littlewood, and Polya, are easily obtained by showing that generalized means and generalized convex functions are in fact ordinary arithmetic means and ordinary convex functions, respectively, defined on linear spaces with suitably chosen operations of addition and multiplication. The results are applied to some problems in statistical decision theory.This research was supported by Project No. NR-047-021, Contract No. N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas, Austin, Texas, and by NSF Grant No. ENG-76-10260 at Northwestern University, Evanston, Illinois.