New upper bounds for Mathieu-type series

Abstract

The Mathieu's series $S(r)$ was considered firstly by E.L. Mathieu in 1890, its alternating variant $\widetilde{S}(r)$ has been recently introduced by Pogany et al. [Appl. Math. Comput. 173 (2006), 69--108], where various bounds have been established for $S, \widetilde{S}$. In this note we obtain new upper bounds over $S(r), \widetilde{S}(r)$ with the help of Hardy--Hilbert double integral inequality.