On a new generalization of Alzer's inequality
Summary: Let \(\{a_n\}^\infty_{n=1}\) be an increasing sequence of positive real numbers. Under certain conditions on this sequence we use the mathematical induction and the Cauchy mean-value theorem to prove the following inequality: \[ {a_n\over a_{n+ m}}\leq \Biggl({(1/n) \sum^n_{i=1} a^r_i\over (1/(n+ m)) \sum^{n+ m}_{i=1} a^r_i}\Biggr)^{1/r}, \] 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 [\textit{H. Alzer}, J. Math. Anal. Appl. 179, No. 2, 396-402 (1993; Zbl 0792.26008)] in a new direction. It is shown that the above inequality holds for a large class of positive, increasing and logarithmically concave sequences.
scientific article; zbMATH DE number 1505956