  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.

Classical Analysis and ODEs