International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 12, Pages 815-818
doi:10.1155/S0161171200003033

On a new generalization of Alzer's inequality

Feng Qi1 and Lokenath Debnath2

1Department of Mathematics, Jiaozuo Institute of Technology, Henan, Jiaozuo City 454000, China
2Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA

Received 2 April 1999; Revised 10 December 1999

Copyright © 2000 Feng Qi and Lokenath Debnath. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let {an}n=1∞ be an increasing sequence of positive real numbers. Under certain conditions of this sequence we use the mathematical induction and the Cauchy mean-value theorem to prove the following inequality: anan+m≤((1/n)∑i=1nair(1/(n+m))∑i=1n+mair)1/r, where n and m are natural numbers and r is a positive number. The lower bound is best possible. This inequality generalizes the Alzer's inequality (1993) in a new direction. It is shown that the above inequality holds for a large class of positive, increasing and logarithmically concave sequences.