Note on an inequality for infinite series (Q1911350)

The main result of this paper is Theorem 1: Let \(M>0\) and \(a_k>0\) \((k=0,1,2,\dots)\) be real numbers. If \[ A_{n+1}= \sum^\infty_{k=n+1} a_k\leq M\cdot a_n\qquad (n=0,1,2,\dots) \] then we have for all \(c\in (0,1)\): \[ c\sum^\infty_{n=0} (M\cdot a_n-A_{n+1}) (A^{c-1}_{n+1}-A_n^{c-1})+ [(M+1)^c-M^c] \sum^\infty_{n=0}a^c_n\leq A^c_0. \] As an application, an upper bound for entropy of a probability distribution is derived. These results give refinements of two theorems proved by \textit{J.-P. Alouche}, \textit{M. Mendès France} and \textit{G. Tenenbaum} [Tokyo J. Math. 11, No. 2, 323-328 (1988; Zbl 0685.28010)].