A general form of Alzer's inequality (Q1886474)

Alzer's inequality from 1993 states that \(S_n(r)/S_{n+1}(r)\geq (n/(n+ 1))^r\) for \(n= 1,2,\dots\); \(r> 0\), where \(S_n(r)= {1\over n}\sum^n_{i=1} i^r\). The first easy proof for Alzer's inequality has been obtained in 1995 by the reviewer, who used mathematical induction and Cauchy's mean value theorem of differential calculus. The authors now use this method to deduce generalizations for sequences of positive integers, satisfying certain auxiliary conditions.