An invariance formula in the class of generalized p-variable quasiarithmetic means is provided. An effective form of the limit of the sequence of iterates of mean-type mappings of this type is given. An application to determining functions which are invariant with respect to generalized quasiarithmetic mean-type mappings is presented.

Under the assumption of twice continuous differentiability of some of the functions involved we determine all the weighted quasi-arithmetic means M,N,K such that K is (M,N)-invariant, that is, K∘(M,N) = K. Some applications to iteration theory and functional equations are presented.

Let I be a real interval, J a subinterval of I, p ≥ 2 an integer number, and M1, ... , Mp : Ip → I the continuous means. We consider the problem of invariance of the graphs of functions ϕ : Jp−1 → I with respect to the mean-type mapping M = (M1, ... , Mp).Applying a result on the existence and uniqueness of an M -invariant mean [7], we prove that if the graph of a continuous function ϕ : Jp−1 → I ...

We work with a fixed N-tuple of quasi-arithmetic means $M₁,...,M_N$ generated by an N-tuple of continuous monotone functions $f₁,...,f_N:I\to ℝ$ (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping $I^N\ni b\mapsto (M₁\left(b\right),...,M_N\left(b\right))$ tend pointwise to a mapping having values on the diagonal of $I^N$. Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M₁,...,M_N$ taken on b. We effectively...

Several mean value theorems for higher order divided differences and approximate Peano derivatives are proved.

We analyse mean values of functions with values in the boundary of a convex two-dimensional set. As an application, reverse integral inequalities imply exactly the same inequalities for the monotone rearrangement. Sharp versions of the classical Gehring lemma, the Gurov-Resetnyak theorem and the Muckenhoupt theorem are obtained.

For a differentiable function $𝐟:I\to {ℝ}^k,$ where $I$ is a real interval and $k\in \mathbb{N}$, a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean $M:I^2\to I$ such that$$𝐟\left(x\right)-𝐟\left(y\right)=(x-y){𝐟}^{\text{'}}\left(M(x,y)\right),x,y\in I,$$ are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.

The author gives a new simple proof of monotonicity of the generalized extended mean values $M(r,s)=\le {(\left(\int f^{s}d\mu \right)/\left(\int f^{r}d\mu \right))}^{1/(s-r)}$ introduced by F. Qi.

In this work we introduce a nonparametric recursive aggregation process called Multilayer Aggregation (MLA). The name refers to the fact that at each step the results from the previous one are aggregated and thus, before the final result is derived, the initial values are subjected to several layers of aggregation. Most of the conventional aggregation operators, as for instance weighted mean, combine numerical values according to a vector of weights (parameters). Alternatively, the MLA operators...