International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2000 / Article

Open Access

Volume 23 |Article ID 698020 | https://doi.org/10.1155/S0161171200003033

Feng Qi, Lokenath Debnath, "On a new generalization of Alzer's inequality", International Journal of Mathematics and Mathematical Sciences, vol. 23, Article ID 698020, 4 pages, 2000. https://doi.org/10.1155/S0161171200003033

On a new generalization of Alzer's inequality

Received02 Apr 1999
Revised10 Dec 1999

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.

Copyright © 2000 Hindawi Publishing Corporation. 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.


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