On the balancing property of Matkowski means

Let $$I\subseteq \mathbb {R}$$ I ⊆ R be a nonempty open subinterval. We say that a two-variable mean $$M:I\times I\rightarrow \mathbb {R}$$ M : I × I → R enjoys the balancing property if, for all $$x,y\in I$$ x , y ∈ I , the equality $$\begin{aligned} {M\big (M(x,M(x,y)),M(M(x,y),y)\big )=M(x,y)} \end{aligned}$$ M ( M ( x , M ( x , y ) ) , M ( M ( x , y ) , y ) ) = M ( x , y ) holds. The above equation has been investigated by several authors. The first remarkable step was made by Georg Aumann in 1935. Assuming, among other things, that M is analytic, he solved (1) and obtained quasi-arithmetic means as solutions. Then, two years later, he proved that (1) characterizes regular quasi-arithmetic means among Cauchy means, where, the differentiability assumption appears naturally. In 2015, Lucio R. Berrone, investigating a more general equation, having symmetry and strict monotonicity, proved that the general solutions are quasi-arithmetic means, provided that the means in question are continuously differentiable. The aim of this paper is to solve (1), without differentiability assumptions in a class of two-variable means, which contains the class of Matkowski means.

On the equality of two-variable general functional means

Given two functions $$f,g:I\rightarrow \mathbb {R}$$ f , g : I → R and a probability measure $$\mu $$ μ on the Borel subsets of [0, 1], the two-variable mean $$M_{f,g;\mu }:I^2\rightarrow I$$ M f , g ; μ : I 2 → I is defined by $$\begin{aligned} M_{f,g;\mu }(x,y) :=\bigg (\frac{f}{g}\bigg )^{-1}\left( \frac{\int _0^1 f\big (tx+(1-t)y\big )d\mu (t)}{\int _0^1 g\big (tx+(1-t)y\big )d\mu (t)}\right) \quad (x,y\in I). \end{aligned}$$ M f , g ; μ ( x , y ) : = ( f g ) - 1 ∫ 0 1 f ( t x + ( 1 - t ) y ) d μ ( t ) ∫ 0 1 g ( t x + ( 1 - t ) y ) d μ ( t ) ( x , y ∈ I ) . This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević means. The aim of this paper is, for a fixed probability measure $$\mu $$ μ , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which $$\begin{aligned} M_{f,g;\mu }(x,y)=M_{F,G;\mu }(x,y) \quad (x,y\in I) \end{aligned}$$ M f , g ; μ ( x , y ) = M F , G ; μ ( x , y ) ( x , y ∈ I ) holds. Under at most sixth-order differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarević means and of Cauchy means.

Multidimensional reversed Hardy type inequalities for monotone functions

In this paper, we will give some multidimensional generalization of reversed Hardy type inequalities for monotone functions. Moreover, we will give n-exponential convexity, exponential convexity and related results for some functionals obtained from the differences of these inequalities. At the end we will give mean value theorems and Cauchy means for these functionals.

Mean-value theorems and some symmetric means

Some variants of the Lagrange and Cauchy mean-value theorems lead to the conclusion that means, in general, are not symmetric. They are symmetric iff they coincide (respectively) with the Lagrange and Cauchy means. Under some regularity assumptions, we determine the form of all the relevant symmetric means.