Some inequalities involving generalized Bessel functions

Let up be the generalized and normalized Bessel function depending on parameters b, c, p and let lambda(p) (x) = up (x(2)), x epsilon R. In this paper we extend to the function lambda(p) some wellknown classical inequalities like Mahajan's inequality, Mitrinovic's inequality, improvements of Jordan's inequality, Redheffer's inequality, using an adequate integral representation of the function lambda(p) and the monotone form of l'Hospital's rule. Moreover we prove that the integral, sigma(p) (x) = integral(x)(0) lambda(p) (t) dt is sub-additive (super-additive) under certain conditions on parameters b, c, p.