论文标题
关于Matkowski的平衡属性意味着
On the balancing property of Matkowski means
论文作者
论文摘要
令$ i \ subseteq \ mathbb {r} $为非空的subinterval。我们说，如果，对于所有$ x，y \ in i $，equality \ begin {equation} \ tag {1} m \ big（m（x，m（x，y）），m（m（x，y），y）\ big）= m（x，y）\ end {equation {equation}保持。 上述方程已由几位作者进行了研究。 Georg Aumann于1935年迈出了一个显着的步骤。除其他外，他解决了（1），并获得了准算术手段作为解决方案。然后，两年后，他证明了（1）表征了\ emph {常规} Quasi-arithmetic Menetic Menecenter Mene之间，其中，不同的性能自然而然。 2015年，Lucio R. Berrone研究了一个更通用的方程式，具有对称性和严格的单调性，证明了通用解决方案是准算术手段，但前提是所涉及的手段是\ emph {连续可区分的}。 本文的目的是解决（1），而在一类两种变量的手段中没有可区分性假设，其中包含\ emph {matkowski Mene含义}的类别。
Let $I\subseteq\mathbb{R}$ be a nonempty open subinterval. We say that a two-variable mean $M:I\times I\to\mathbb{R}$ enjoys the \emph{balancing property} if, for all $x,y\in I$, the equality \begin{equation}\tag{1} M\big(M(x,M(x,y)),M(M(x,y),y)\big)=M(x,y) \end{equation} 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 \emph{analytic}, he solved (1) and obtained quasi-arithmetic means as solutions. Then, two years later, he proved that (1) characterizes \emph{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 \emph{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 \emph{Matkowski means}.
