On some inequalities of the Bateman's G-function.

In the paper, we prove that the Bateman's G--function satisfies the double inequality Σ2m n=1 (2n-1)B2n/nx2n < G(x) - 1/x < Σ2m-1 n=1 (2n-1)B2n/nx2n , m ∊ N with best bounds, where B'rs 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. [ABSTRACT FROM AUTHOR]