Inequalities for some infinite series (Q1297806)

The author proves that if \(M\) is a positive real number and \(a_n>0\) \((n\in{\mathbb N})\) is a sequence such that \(r_{n+1}:=\sum_{k=n+1}^{\infty}a_k\leq Ma_n\) then, for an arbitrary \(c\in{}]0,1[\), the series \(\sum_{n=1}^{\infty}a_n^c\) and \(\sum_{n=1}^{\infty}r_{n+1}a_n^{c-1}\) are convergent, furthermore, \[ [(M+1)^c-M^c]\sum_{n=1}^{\infty}a_n^c +c[M^{c-1}-(M+1)^{c-1}] \sum_{n=1}^{\infty}(Ma_n-r_{n+1})a_n^{c-1} \leq\biggl(\sum_{n=1}^{\infty}a_n\biggr)^c. \] As an application of this theorem an inequality concerning the Shannon entropy of infinite probability distributions is shown. These statements are refinements of the results by \textit{J.-P. Allouche, M. Mendès France} and \textit{G. Tenenbaum} [Tokyo J. Math. 11, No. 2, 323-328 (1988; Zbl 0685.28010)], which were generalized incorrectly by \textit{H. Alzer} [Acta Math. Hung. 67, No. 3, 203-206 (1995; Zbl 0866.26010)].