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
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.