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.