Extension of an Inequality of H. Alzer for Negative Powers

Abstract

In this paper, we show that let $n$ be a natural number, then for all real numbers $ r $,

$$ \frac{n}{n+1}<\left(\frac{1}{n}\sum_{i=1}^{n}i^{r} \Big / \frac{1}{n+1}\sum_{i=1}^{n+1}i^{r}\right)^{1/r}<1.$$

Both bounds are best possible. This extends a result of H. Alzer, who established this inequality for $ r>0 $.