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Bounds for Bateman’s \(G\)-function and its applications. (English) Zbl 1351.33003
Summary: In this paper, we present an asymptotic formula for Bateman’s \(G\)-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 \(\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k+h}\), \(h\neq-1,-2,-3,\dots\). 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.

33B15 Gamma, beta and polygamma functions
26D15 Inequalities for sums, series and integrals
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41A80 Remainders in approximation formulas
AMath; DAmath; DLMF; Equator
Full Text: DOI
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