Toader型平均关于其他二元平均的逼近

On approximation the Toader-type mean by other bivariate means

通过研究高斯算术几何平均和其他平均的组合或复合函数的单调性,给出了Toader型平均值的由算术几何平均给出的几个精确逼近。作为所得结果的应用,利用Toader型平均值与第二类完全椭圆积分的关系,得到第二类完全椭圆积分的新的上下界。这些上下界提供了第一类和第二类完全椭圆积分之间的联系。

In the article,the authors provide several sharp approximations for the Toader-type mean in terms of Gaussian arithmetic-geometric mean by showing the monotonicity properties for the combinations or compositions of these related means.As the application of the results,by use of the relationship between Toader-type mean and the complete elliptic integral of the second kind,the new bounds for the complete elliptic integral of the second kind were obtained,which provided the relations between the complete integrals of the first kind and second kind.