The Mathieu’s series S(r) was considered firstly by É.L. Mathieu in 1890;
its alternating variant Š(r) has been recently introduced by Pogány et al. [Some families of Mathieu a-series and
alternating Mathieu a-series, 2006]
where various bounds have been established for S, Š. In this note we obtain
new upper bounds over S(r), Š(r) with the help of Hardy–Hilbert double integral
inequality.