Extension of an Inequality of H. Alzer for Negative Powers

Authors

  • Chao-Ping Chen
  • Feng Qi

DOI:

https://doi.org/10.5556/j.tkjm.36.2005.137

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 $.

Author Biographies

Chao-Ping Chen

Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, #142, Mid-Jiefang Road, Jiaozuo City, Henan 454000, China.

Feng Qi

Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, #142, Mid-Jiefang Road, Jiaozuo City, Henan 454000, China.

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Published

2005-03-31

How to Cite

Chen, C.-P., & Qi, F. (2005). Extension of an Inequality of H. Alzer for Negative Powers. Tamkang Journal of Mathematics, 36(1), 69–72. https://doi.org/10.5556/j.tkjm.36.2005.137

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Section

Papers

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