Symmetric Functions and the Theorem of the Arithmetic and Geometric Means

We show that results analogous to the theorem of the arithmetic and geometric means hold for the three multiplicative fundamental bases of the vector space of symmetric functions – the elementary symmetric functions, the homogeneous symmetric functions, and the power sum symmetric functions. We give examples to demonstrate that no such results hold for the two non-multiplicative fundamental bases – the Schur functions and the monomial symmetric functions.