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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
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[1] Andrews G. E., Askey R. and Roy R., Special Functions, Encyclopedia Math. Appl. 71, Cambridge University Press, Cambridge, 1999.
[2] Copson E. T., Asymptotic Expansions, Cambridge Tracts in Math. Math. Phys. 55, Cambridge University Press, New York, 1965.
[3] Ehrhardt W., The AMath and DAMath special functions: Reference manual and implementation notes, version 2.03, 2014, .
[4] Erdélyi A., Higher Transcendental Functions. Vols. I-III. Based on Notes Left by Harry Bateman, McGraw-Hill Book Company, New York, 1953-1955.
[5] Guo B.-N. and Qi F., Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin) 34 (2014), no. 2, 201-208. · Zbl 1294.33003
[6] Kazarinoff N. D., Analytic Inequalities, Holt, Rinehart and Winston, New York, 1961. · Zbl 0097.03801
[7] Koumandos S., Monotonicity of some functions involving the psi function and sharp estimates for the remainders of certain series, Int. J. Contemp. Math. Sci. 2 (2007), no. 13-16, 713-720. · Zbl 1131.33001
[8] Kuang J.-C., Applied Inequalities (in Chinese), 3rd ed., Shandong Science and Technology Press, Jinan City, 2004.
[9] Mortici C., A sharp inequality involving the psi function, Acta Univ. Apulensis Math. Inform. 21 (2010), 41-45. · Zbl 1212.33002
[10] Mortici C., Estimating the digamma and trigamma functions by completely monotonicity arguments, Appl. Math. Comput. 217 (2010), no. 8, 4081-4085. · Zbl 1207.33002
[11] Mortici C., New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett. 23 (2010), no. 1, 97-100. · Zbl 1183.33003
[12] Mortici C., Accurate estimates of the gamma function involving the PSI function, Numer. Funct. Anal. Optim. 32 (2011), no. 4, 469-476. · Zbl 1226.33001
[13] Mortici C., The quotient of gamma functions by the psi function, Comput. Appl. Math. 30 (2011), no. 3, 627-638. · Zbl 1247.33007
[14] Mortici C. and Chen C.-P., New sharp double inequalities for bounding the gamma and digamma function, An. Univ. Vest Timiş. Ser. Mat.-Inform. 49 (2011), no. 2, 69-75. · Zbl 1274.30149
[15] Oldham K., Myland J. and Spanier J., An Atlas of Functions. With Equator, the Atlas Function Calculator, 2nd ed., Springer, New York, 2009. · Zbl 1167.65001
[16] Olver F. W. J., Lozier D. W., Boisvert R. F. and Clark C. W., NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010. · Zbl 1198.00002
[17] Qi F., Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities, Filomat 27 (2013), no. 4, 601-604. · Zbl 1324.33004
[18] Qi F., Cerone P. and Dragomir S. S., Complete monotonicity of a function involving the divided difference of PSI functions, Bull. Aust. Math. Soc. 88 (2013), no. 2, 309-319. · Zbl 1280.26018
[19] Qi F., Chen S.-X. and Cheung W.-S., Logarithmically completely monotonic functions concerning gamma and digamma functions, Integral Transforms Spec. Funct. 18 (2007), no. 5-6, 435-443. · Zbl 1120.33003
[20] Qi F. and Guo B.-N., Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2014), no. 1, 63-72.
[21] Qi F. and Guo B.-N., Complete monotonicity of divided differences of the di- and tri-gamma functions with applications, Georgian Math. J. 23 (2016), no. 2, 279-291. · Zbl 1339.33007
[22] Qi F., Niu D.-W. and Guo B.-N., Refinements, generalizations, and applications of Jordan’s inequality and related problems, J. Inequal. Appl. 2009 (2009), Article ID 271923. · Zbl 1175.26048
[23] Qiu S.-L. and Vuorinen M., Some properties of the gamma and psi functions, with applications, Math. Comp. 74 (2005), no. 250, 723-742. · Zbl 1060.33006
[24] Sîntămărian A., A new proof for estimating the remainder of the alternating harmonic series, Creat. Math. Inform. 21 (2012), no. 2, 221-225. · Zbl 1289.40012
[25] Sîntămărian A., Sharp estimates regarding the remainder of the alternating harmonic series, Math. Inequal. Appl. 18 (2015), no. 1, 347-352. · Zbl 1401.11162
[26] Timofte V., On Leibniz series defined by convex functions, J. Math. Anal. Appl. 300 (2004), no. 1, 160-171. · Zbl 1058.40002
[27] Timofte V., Integral estimates for convergent positive series, J. Math. Anal. Appl. 303 (2005), no. 1, 90-102. · Zbl 1069.40002
[28] Tóth L., On a class of Leibniz series, Rev. Anal. Numér. Théor. Approx. 21 (1992), no. 2, 195-199. · Zbl 0801.40004
[29] Tóth L. and Bukor J., On the alternating series \({1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+⋯}\), J. Math. Anal. Appl. 282 (2003), no. 1, 21-25. · Zbl 1025.40001
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