Aequationes Mathematicae ( IF 0.9 ) Pub Date : 2020-09-30 , DOI: 10.1007/s00010-020-00755-w László Losonczi , Zsolt Páles , Amr Zakaria
Given two functions \(f,g:I\rightarrow \mathbb {R}\) and a probability measure \(\mu \) on the Borel subsets of [0, 1], the two-variable mean \(M_{f,g;\mu }:I^2\rightarrow I\) is defined by
$$\begin{aligned} M_{f,g;\mu }(x,y) :=\bigg (\frac{f}{g}\bigg )^{-1}\left( \frac{\int _0^1 f\big (tx+(1-t)y\big )d\mu (t)}{\int _0^1 g\big (tx+(1-t)y\big )d\mu (t)}\right) \quad (x,y\in I). \end{aligned}$$This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević means. The aim of this paper is, for a fixed probability measure \(\mu \), to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which
$$\begin{aligned} M_{f,g;\mu }(x,y)=M_{F,G;\mu }(x,y) \quad (x,y\in I) \end{aligned}$$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.
中文翻译：
关于二变量一般函数均值的相等性
给定两个函数\（f，g：I \ rightarrow \ mathbb {R} \）和概率度量\（\ mu \）的[0，1]的Borel子集，两个变量的均值\（M_ {f ，g; \ mu}：I ^ 2 \ rightarrow I \）由
$$ \ begin {aligned} M_ {f，g; \ mu}（x，y）：= \ bigg（\ frac {f} {g} \ bigg）^ {-1} \ left（\ frac {\ int _0 ^ 1 f \ big（tx +（1-t）y \ big）d \ mu（t）} {\ int _0 ^ 1 g \ big（tx +（1-t）y \ big）d \ mu（t） } \ right）\ quad（x，y \ in I）。\ end {aligned} $$此类手段包括拟算术以及柯西和Bajraktarević手段。本文的目的是，对于一个固定的概率测度\（\ mu \），研究它们的相等性问题，即表征那些对（f， g）和（F， G）的函数对。
$$ \ begin {aligned} M_ {f，g; \ mu}（x，y）= M_ {F，G; \ mu} {x，y）\ quad（x，y \ in I）\ end {aligned } $$持有。在未知函数f， g和F， G的最多六阶可微性假设下，我们用常微分方程式为上述方程的解提供了几个必要条件。对于两个特定的措施，获得了完整的描述。后面的结果为Bajraktarević均值和Cauchy均值的相等提供了八个等效条件。
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