International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 12, Pages 815-818
On a new generalization of Alzer's inequality
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.
Let 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:
, where and are natural numbers and 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.