Invariant Means Alan Horwitz Penn State University 25

Let m(a, b) and M(a, b, c) be symmetric means. We say that M is type 1 invariant with respect to m if M(m(a, c),m(a, b),m(b, c)) ≡ M(a, b, c). If m is strict and isotone, then we show that there exists a unique M which is type 1 invariant with respect to m. In particular we discuss the invariant logarithmic mean L3, which is type 1 invariant with respect to L(a, b) = b−a log b−log a . We say that M is type 2 invariant with respect to m if M(a, b,m(a, b)) ≡ m(a, b). We also prove existence and uniqueness results for type 2 invariance, given the mean M(a, b, c). The arithmetic, geometric, and harmonic means in two and three variables satisfy both type 1 and type 2 invariance. There are means m and M such that M is type 2 invariant with respect to m, but not type 1 invariant with respect to m(for example, the Lehmer means). L3 is type 1 invariant with respect to L, but not type 2 invariant with respect to L.