On weak monotonicity of some mixture functions

Monotonicity with respect to all arguments is fundamental to the definition of aggregation functions. Here we study means that are not necessarily monotone. Weak monotonicity was recently proposed as a relaxation of the monotonicity condition for averaging functions. We provide results for the weak monotonicity of some importantclasses of mixture functions. With these results we are able to extend and improve the understanding of this very important class of functions

Invariance property for partial means

We study the properties of the mean-type mappings ${\bf M}\colon I^p \to I^p$ of the form $${\bf M}(x_1,\dots,x_p):=\big(M_1(x_{\alpha_{1,1}},\dots,x_{\alpha_{1,d_1}}),\dots,M_p(x_{\alpha_{p,1}},\dots,x_{\alpha_{p,d_p}})\big),$$ where $p$ and $d_i$-s are positive integers, each $M_i$ is a $d_i$-variable mean on an interval $I \subset \mathbb{R}$, and $\alpha_{i,j}$-s are elements from $\{1,\dots,p\}$. We show that, under some natural assumption on $M_i$-s, the problem of existing the unique $\bf M$-invariant mean can be reduced to the ergodicity of the directed graph with vertexes $\{1,\dots,p\}$ and edges $\{(\alpha_{i,j},i) \colon i,j \text{ admissible}\}$

Weak monotonicity of Lehmer and Gini means

We analyze directional monotonicity of several mixture functions in the direction (1,1...,1), called weak monotonicity. Our particular focus is on power weighting functions and the special cases of Lehmer and Gini means. We establish limits on the number of arguments of these means for which they are weakly monotone. These bounds significantly improve the earlier results and hence increase the range of applicability of Gini and Lehmer means. We also discuss the case of affine weighting functions and find the smallest constant which ensures weak monotonicity of such mixture functions