Countable Mean Family With Multiple Parameters In Higher Space

Mean functions are a very important part in the theory of mean values. In paper [1][2] and [3] , mean functions were axiomatized and the mean family theory was built up. In this paper, based on paper [1] [2]and [3] , countable mean family with multiple parameters is discussed concretely and deeply.This paper is composed of four parts. The first part is the introduction, in which we simply introduce some results of the related mean functions. In chapter 1, firstly, some preparation theorems and the theoretical ground in which countable mean family is defined in closed variable region are given. Then, countable mean functionals are defind and the operation relationship of them is studied . Finally, countable mean family is defined and two specific countable mean families: countable power mean family and contable exponential mean family are given,meanwhile, their properties such as their inequalities,their mononicity as functionals of variable group a and so on are discussed. In chapter 2, we introduce two types multi-parameter countable mean families from different angles: the one type is two-parameter power and exponential countable mean family, the other type is arithmetical,geometric and harmonic formal countable mean family with multiple prarameters, and from several respects,their properties are studied. In the last part,basing on the countable mean family on one-dimension space, countable mean family on n-dimention space is built up and some correlated analysis is made simply.The theory in this paper prepares for its extensive application in numerical approximation and inequalities.