Lüroth Expansion Digits and Maclaurin’s Inequality

作     者：LI Li CAO Fang TANG Shixin WU Yuhan 无

作者机构：School of Mathematics and Statistics Wuhan UniversityWuhan 430072 Hubei China The True Light Middle School in Guangzhou Guangzhou 510145 Guangdong China 

出 版 物：《Wuhan University Journal of Natural Sciences》 (武汉大学学报（自然科学英文版）)

年 卷 期：2018年第23卷第6期

页      面：471-474页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：Supported by the National Natural Science Foundation of China(11271148) 

主　　题：Luroth series expansion Maclaurin's inequalities arithmetic mean geometric mean 

摘      要：It is well known that for almost all real number x, the geometric mean of the first n digits di（x） in the Lüroth expansion of x converges to a number K0 as n→∞. On the other hand, for almost all x, the arithmetric mean of the first n Lüroth expansion digits di（x） approaches infinity as n→∞. There is a sequence of refinements of the AM-GM inequality, Maclaurin＇s inequalities, relating the 1/k-th powers of the k-th elementary symmetric means of n numbers for 1≤k≤n. In this paper, we investigate what happens to the means of Lüroth expansion digits in the limit as one moves f（n） steps away from either extreme. We prove sufficient conditions on f（n） to ensure divergence when one moves away from the arithmetic mean and convergence when one moves f（n） steps away from geometric mean.