Feng Qi
An Integral Expression and Some Inequalities of Mathieu Type Series
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. [58], [37]-[46]([2004])
MSC:
26D15 Inequalities for sums, series and integrals
33E20 Other functions defined by series and integrals
Abstract: Let $r>0$ and $a=\{a_k>0,k\in\mathbb{N}\}$ such that the series
$g(x)=\sum_{k=1}^\infty e^{-a_kx}$ converges for $x>0$,
then the Mathieu type series
$\sum_{k=1}^\infty\frac{a_k}{\left(a_k^2+r^2\right)^2}=\frac1{2r}\int_0^\infty xg(x)\sin(rx)\td x$.\par
If $a=\{a_k>0,k\in\mathbb{N}\}$ is an arithmetic sequence,
then some inequalities of Mathieu type series
$\sum_{k=1}^\infty\frac{a_k}{\left(a_k^2+r^2\right)^2}$ are obtained for $r>0$.

Keywords: Mathieu type series, integral expression, Laplace transform, inequality