均值及均值方程

Means and the Equations of Means

均值及均值方程是函数方程领域近二十年研究的热门方向.本文首先介绍了均值的定义及性质、Gauss迭代、不变方程和M-S问题的发展历程.接着围绕Bajraktarevic均值、Cauchy均值及其他相关均值进行综述,尤其详述了对应的相等性与不变性问题的研究进展.Bajraktarevic均值和Cauchy均值这两类均值均含两个衍生函数,对应的等式问题已基本得以解决,但这两类对称均值的不变性因含有四个未知函数而求解困难的问题目前仍未完全解决.最后介绍了均值在其他领域中的应用.

Means and the equations of means are one of the most influential topics of functional equations during the last two decades.We firstly introduce the definitions and properties of means,Gauss iteration,invariance equations and the development history of M-S problem.Then we give the survey focusing on Bajraktarevic and Cauchy means,also on some other surroundings,especially present research progress of equality and invariance of them in details.The equality involving Bajraktarevic and Cauchy means which have two derived functions have almost been solved,but it is difficult to solve the invariance of these two types of means,which contains four unknown functions.There still exist many open problems.Finally,the application of means in other fields has been introduced.