Generalization of an inequality of Alzer for negative powers

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.