Generalization of an inequality of Alzer for negative powers

Main Article Content

Chao-Ping Chen
Feng Qi

Abstract

Let $ \{a_n \}_{n=1}^{\infty} $ be a positive, strictly increasing, and logarithmically concave sequence satisfying $ (a_{n+1}/a_{n})^{n}<(a_{n+2}/a_{n+1})^{n+1} $. Then we have

$$\frac {a_n}{a_{n+m}}<\left(\frac {1}{n} \sum_{i=1}^{n}a_{i}^r\bigg/ \frac {1}{n+m} \sum_{i=1}^{n+m}a_{i}^r \right)^{1/r}, $$

where $ n $, $ m $ are natural numbers and $ r $ is a positive real number. The lower bound is the best possible. This generalizes an inequality of Alzer for negative powers.

Article Details

How to Cite
Chen, C.-P., & Qi, F. (2005). Generalization of an inequality of Alzer for negative powers. Tamkang Journal of Mathematics, 36(3), 219–222. https://doi.org/10.5556/j.tkjm.36.2005.113
Section
Papers
Author Biographies

Chao-Ping Chen

Department of Applied Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan 454010, CHINA.

Feng Qi

Department of Applied Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan 454010, CHINA.