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

26A42 Integrals of Riemann, Stieltjes and Lebesgue type, See also {28-XX}

40A30 Convergence and divergence of series and sequences of functions

44A10 Laplace transform

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
Notes: The author was supported in part by NNSF of China, SF for the Prominent Youth of
Henan Province, SF of Henan Innovation Talents at Universities, Doctor Fund of Jiaozuo Institute of
Technology, CHINA