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Let $I\subset (0,\infty )$ be an interval that is closed with respect to themultiplication. 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 generalizedweighted quasi-geometric means is presented. It is shown that invariancequestion within the class of this operations leads to means of iterative typeand to a problem on a composite functional equation. An application of theinvariance identity to determine effectively the limit of the sequence ofiterates of some generalized quasi-geometric mean-type mapping, and the form ofall continuous functions which are invariant with respect to this mapping aregiven. The equality of two considered operations is also discussed.

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