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.
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.
基金
Supported by the National Natural Science Foundation of China(11271148)
