  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.
