双纽线平均的算术与二次平均调和组合界

Sharp Harmonic Combination Bounds for Lemniscatic Means in Terms of Arithmetic and Quadratic Means

双纽线平均是一种迭代平均,它可以被高斯反双纽线正弦、反双纽线双曲正弦等双纽线函数表示出来.双纽线平均复合不同阶的幂平均可得到一些衍生平均.研究双纽线平均及其衍生平均与不同阶幂平均的各种特殊组合的序关系,可得到一些有价值的平均值不等式.本文运用实分析的方法,研究了双纽线平均和算术与二次平均调和组合的序关系,得到了有关双纽线平均的两个最佳双向不等式,从而推得反双纽线双曲正弦函数和反双纽线正切函数新的确界.

The lemniscate mean is an iterative mean,which can be expressed in terms of some lemniscate functions,such as Gauss’arc lemniscate sine and the hyperbolic arc lemniscate sine functions,etc.Lemniscate mean compositing power means of different orders can be used to derive some important bivariate means.Studying the order relations of the lemniscate mean and its derived means with a variety of special combinations of the different order power means can get some valuable means inequalities.In this paper,the method of real analysis was used to study the order relations of the lemniscate mean with harmonic combination of arithmetic and quadratic means.The two optimal bidirectional inequalities for lemniscate mean are found,with the estimations of Gauss’arc lemniscate hyperbolic sine and arc lemniscate tangent functions respectively obtained.