Some Inequalities for Geometric Means

Abstract

This paper is mainly concerned with a discrete inequality bearing some resemblance to recent inequalities of Heinig and of Cochran and Lee. These are typified by

$$\sum\limits_{\text{j = 0}}^\text{n} {\text{m}^{\text{k} - 1} \left[ {\prod\limits_{\text{n} = 1}^\text{m} {\text{x}_\text{n}^{\text{n}^{\text{p - 1}} } } } \right]\text{p/m}^\text{p} \leqslant \text{e}^{\text{k/p}} \sum\limits_{\text{m} = 1}^\infty {\text{m}^{\text{k} - 1} \text{x}_\text{m} } }$$

under appropriate conditions. The products on the left are replaced, in this paper, by geometric means with more general weights, and the factors m^k-1 on both sides by factors r^-m for suitably small r. Some inequalities having an analogous character are first discussed, since they led the way into this study.