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See detailInvariance in a class of operations related to weighted quasi-geometric means
Devillet, Jimmy UL; Matkowski, Janusz

E-print/Working paper (2018)

Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left ... [more ▼]

Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left( f\circ g\right) ^{-1}\left( f\left( x\right) \cdot g\left( y\right) \right) \text{,} \end{equation*} 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. [less ▲]

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See detailA new invariance identity and means
Devillet, Jimmy UL; Matkowski, Janusz

in Results in Mathematics (2018), 73(4), 130

The invariance identity involving three operations D_{f,g} : X x X -> X of the form D_{f,g} (x,y) = (f o g)^{-1} (f (x) + g (y)) , is proposed. The connections of these operations with means is ... [more ▼]

The invariance identity involving three operations D_{f,g} : X x X -> X of the form D_{f,g} (x,y) = (f o g)^{-1} (f (x) + g (y)) , is proposed. The connections of these operations with means is investigated. The question when the invariance equality admits three means leads to a com- posite functional equation. Problem to determine its continuous solutions is posed [less ▲]

Detailed reference viewed: 123 (49 UL)