ISSN:
1420-8903
Keywords:
Primary 39B40
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Gini, Lehmer, Beckenbach, and others studied the meanG s (a, b) = (a s +b s )/(a s 1 +b s-1 ) We prove Theorem 1 The identity (ina, b)G s (G t ,G u ) =G v holds if and only if (s, t, u, v) is (s, t, t, t) (the trivial solution) or one of (1, 1 −k, 1 +k, 1), (1/2, 1 −k, k, 1/2), or (0,−k, k, 0) (the exotic solutions,k is any real number) Theorem 2 IfP s (a, b) is the power mean [(a s +b s )/2]1/s , thenP s (P t ,P u ) =P v has only the trivial solution (s, t, u, v) = (s, t, t, t) and the exotic solution (0,t, −t, 0) The family of meansG s (respP s ) includes the classical arithmetic, geometric, and harmonic means
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01836156
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