Bounds for Bateman's G-function and its applications

Abstract

AbstractIn this paper, we present an asymptotic formula for Bateman’sG-function${G(x)}$and deduce the double inequality$\frac{1}{2x^{2}+3/2}<G(x)-\frac{1}{x}<\frac{1}{2x^{2}},\quad x>0.$We apply this result to find estimates for the error term of the alternating series$\sum_{k=1}^{\infty}\kern-2.0pt\frac{(-1)^{k-1}}{k+h}$,$h\kern-1.0pt\neq\kern-1.0pt-1,\kern-0.5pt-2,\kern-0.5pt-3,\ldots\,$. Also, we study the monotonicity of some functions involving the function${G(x)}$. Finally, we propose a sharp double inequality for the function${G(x)}$as a conjecture.