Justyna Jarczyk

A square iterative roots of some mean-type mappings

Abstract We determine all pairs of weighted quasi-arithmetic means being square iterative roots of another pair , that is we find all continuous strictly monotonic functions and parameters such that the equation is satisfied.

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.

Computer assisted solution of an equality problem of mean values

Abstract Given weighted quasi-arithmetic means M , N the problem is when the mean of the form φ - 1 φ ( x ) + φ ( y ) - 1 2 ( φ ( M ( x , y ) ) + φ ( N ( x , y ) ) ) is again quasi-arithmetic. It was answered in [5] by Daroczy and Pales under the assumption that M = N is arithmetic. We investigate the complementary situation where at least one of the weighted arithmetic means M , N is not arithmetic and solve the problem locally, assuming that the generators of means are four times continuously differentiable. For the computation we use the software package Mathematica 4.0.