  In this paper we characterize generalized quasi-arithmetic means, that is
means of the form $M(x_1,...,x_n):=(f_1+...+f_n)^{-1}(f_1(x_1)+...+f_n(x_n))$,
where $f_1,...,f_n:I\to\mathbb{R}$ are strictly increasing and continuous
functions. Our characterization involves the Gauss composition of the cyclic
mean-type mapping induced by $M$ and a generalized bisymmetry equation.
