Invariant means

Letm andM be symmetric means in two and three variables, respectively.We say thatM 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 thatM 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.  2002 Elsevier Science (USA). All rights reserved.