Parametrized means and limit properties of their Gaussian iterations

We propose the notion of a mean depending on a parameter, define Gauss-type iterates of parametrized mean-type mapping and study their limit behaviour. Also a suitable invariance property of the limit function is established. In such a way we obtain an extension of generalized Gaussian algorithm (Theorem 2.5) to the case of a compact set of parameters. In particular, we generalize some well known results of J. Matkowski dealing with iterates of mean-type mappings not depending on parameter.