Abstract
In the paper, we prove that the Bateman's G-function satisfies the double inequality
Sigma(2m)(n=1) (2(n) - 1)B-2n/nx(2n) < G(x) - 1/x < Sigma(2m-1)(n=1) (2(n) - 1)B-2n/nx(2n), m is an element of N
with best bounds, where B-r's are the Bernoulli numbers and we study the monotonicity of some functions involving the function G(x). Also, we present some estimates for the error term of a class of the alternating series, which improve and generalize some recent resutls and we prove the increasing monotonicity of a sequence arising from computation of the intersecting probability between a plane couple and a convex body.