Jordan type inequalities involving the Bessel and modified Bessel functions

In this work, two general forms of Jordanʹs inequalities for the Bessel and modified Bessel functions are established, and proved by using lʹHospitalʹs rule for monotonicity and some properties of the spherical Bessel and the modified spherical Bessel functions of the first kind. The applications of the results above give two new infinite series for the Bessel and modified Bessel functions.