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

作　　者：李少云[1] 王君丽[2] 何晓红[3] 徐会作 LI Shaoyun;WANG Junli;HE Xiaohong;XU Huizuo(Teacher Development Center,Wenzhou Broadcast and TV University,Wenzhou,Zhejiang,325000;School of Adult Education,Taizhou Vocational College of Science&Technology,Taizhou,Zhejiang 318020;Office of Academic Affairs,Quzhou Broadcast and TV University,Quzhou,Zhejiang 324000;School of Mathematics and Information Engineering,Wenzhou University of Technology,Wenzhou,Zhejiang 325000)

机构地区：[1]温州广播电视大学教师教学发展中心,浙江温州325000 [2]台州科技职业学院成人教育学院,浙江台州318020 [3]衢州广播电视大学教务处,浙江衢州324000 [4]温州理工学院数学与信息工程学院,浙江温州325000

出　　处：《绍兴文理学院学报》2021年第8期40-46,101,共8页Journal of Shaoxing University

基　　金：浙江省教育厅一般科研项目“第一和第二类对称积分及其应用”(Y202043179).

摘　　要：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.

关 键 词：反双纽线函数 双纽线平均 算术平均 二次平均 

分 类 号：O174.6[理学—数学]