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

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.