Invariance in a class of operations related to weighted quasi-geometric means

2018/09/26 by Devillet, Jimmy, Matkowski, Janusz
#26A18 #26E60 #39B12 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.10675

Abstract

Let I⊂ (0,∞ ) be an interval that is closed with respect to the multiplication. The operations C_f,g\colon I²→ I of the form C_f,g( x,y) =( f∘ g) ^-1( f( x) ⋅ g( y) ) , where f,g are bijections of I are considered. Their connections with generalized weighted quasi-geometric means is presented. It is shown that invariance question within the class of this operations leads to means of iterative type and to a problem on a composite functional equation. An application of the invariance identity to determine effectively the limit of the sequence of iterates of some generalized quasi-geometric mean-type mapping, and the form of all continuous functions which are invariant with respect to this mapping are given. The equality of two considered operations is also discussed.

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