【摘要】 Gauss型函数方程是指与平均值相联系的一种函数方程.Gauss最早研究的是针对算术平均值与几何平均值的函数方程,后来,Haruki和Rassias运用Gauss原理研究了针对算术平均数和调和平均数的Gauss型函数方程;Toader研究了针对幂平均值的Gauss型函数方程;刘证研究了针对几何平均数与调和平均数的Gauss型函数方程,但始终没有一个能针对这4个平均值的统一的函数方程.该文在总结他们方法的基础上,研究了一个含有4个iφ(i=1,2,3,4)平均值的Gauss型函数方程,并统一推广了针对算术平均值、几何平均值、调和平均值和幂平均值的相应结果.更多还原

【Abstract】 A Gauss type functional equation is related to means.Gauss initially discussed a functional equation in terms of arithmetic mean and geometric mean.Haruki and Rassias studied a Gauss type functional equation with arithmetic mean and harmonic mean using the Gaussian principle.Toader presented a Gauss type functional equation with power mean.Liu Zheng discussed a functional equation in terms of geometric mean and harmonic mean.However,none of these are associated with all the four means.This paper discusses a Gauss type functional equation having four φ_i(i=1,2,3,4) means,and generalize the results of arithmetic mean,geometric mean,harmonic mean,and power mean.更多还原