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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.

##### MSC:
 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
##### Software:
AMath; DAmath; DLMF; Equator
Full Text:
##### References:
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