Invariant Means
| dc.creator | Horwitz, Alan | |
| dc.date | 2000-07-15 | |
| dc.date.accessioned | 2026-07-07T04:36:24Z | |
| dc.date.available | 2026-07-07T04:36:24Z | |
| dc.description | 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) for all a, b, c > 0. 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 L_3, 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) for all a, b > 0. 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). L_3 is type 1 invariant with respect to L, but not type 2 invariant with respect to L. | |
| dc.description | Submitted for publication-20 pages. No figures | |
| dc.identifier | https://arxiv.org/abs/math/0007095 | |
| dc.identifier | http://arxiv.org/abs/math/0007095 | |
| dc.identifier | Journal of Mathematical Analysis and Applications, 270(2002), 499-518. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/59580 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 26E60 | |
| dc.title | Invariant Means | |
| dc.type | text |