Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function

References

[1] Adell, J. A.—Lekuona, A.: Dirichlet’s eta and beta functions: concavity and fast computation of their derivatives, J. Number Theory 157 (2015), 215–222.10.1016/j.jnt.2015.05.006Search in Google Scholar

[2] Alzer, H.—Kwong, M. K.: On the concavity of Dirichlet’s eta function and related functional inequalities, J. Number Theory 151 (2015), 172–196.10.1016/j.jnt.2014.12.009Search in Google Scholar

[3] Bao, Z.-H.—Agarwal, R. P.—Qi, F.,—Du,W.-S.: Some properties on normalized tails of Maclaurin power series expansion of exponential function, Symmetry 16(8) (2024), Art. 989.10.3390/sym16080989Search in Google Scholar

[4] Bateman, P. T.—Diamond, H. G.: Analytic Number Theory. An Introductory Course. Monographs in Number Theory, Vol. 1. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004.10.1142/5605Search in Google Scholar

[5] Bernstein, D. S.: Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas, Revised and expanded edition, Princeton University Press, Princeton, NJ, 2018.10.1515/9781400888252Search in Google Scholar

[6] Biernacki, M.—Krzyż, J.: On the monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-SkƗodowska Sect. A 9 (1955), 135–147.Search in Google Scholar

[7] Brychkov, Y. A.: Power expansions of powers of trigonometric functions and series containing Bernoulli and Euler polynomials, Integral Transforms Spec. Funct. 20(11-12) (2009), 797–804.10.1080/10652460902867718Search in Google Scholar

[8] Gradshteyn, I. S.—Ryzhik, I. M.: Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, 8th edition, Revised from the 7th edition, Elsevier/Academic Press, Amsterdam, 2015.Search in Google Scholar

[9] Guo, B.-N.—Qi, F.: The function (b^x − a^x/x: Logarithmic convexity and applications to extended mean values, Filomat 25(4) (2011), 63–73.10.2298/FIL1104063GSearch in Google Scholar

[10] Li, W.-H.—Lim, D.—Qi, F.: Expanding the function ln(1+e^x into power series in terms of the Dirichlet eta function and the Stirling numbers of the second kind, Carpathian Math. Publ. 16(1) (2024), 320–327.10.15330/cmp.16.1.320-327Search in Google Scholar

[11] Li, Y.-F.—Qi, F.: A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine, Open Math. 21(1) (2023), Art. 20230159.10.1515/math-2023-0159Search in Google Scholar

[12] Li, Y.-W.—Qi, F.: A new closed-form formula of the Gauss hypergeometric function at specific arguments, Axioms 13(5) (2024), Art. 317.10.3390/axioms13050317Search in Google Scholar

[13] Li, Y.-W.—Qi, F.—Du, W.-S.: Two forms for Maclaurin power series expansion of logarithmic expression involving tangent function, Symmetry 15(9) (2023), Art. 1686.10.3390/sym15091686Search in Google Scholar

[14] Lim, D.—Qi, F.: Increasing property and logarithmic convexity of two functions involving Dirichlet eta function, J. Math. Inequal. 16(2) (2022), 463–469.10.7153/jmi-2022-16-33Search in Google Scholar

[15] Liu, X.-L.—Long, H.-X.—Qi, F.: A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing sine, Mathematics 11(14) (2023), Art. 3107.10.3390/math11143107Search in Google Scholar

[16] Niu, D.-W.—Qi, F.: Monotonicity results of ratios between normalized tails of Maclaurin power series expansions of sine and cosine, Mathematics 12(12) (2024), Art. 1781.10.3390/math12121781Search in Google Scholar

[17] Pei, W.-J.—Guo, B.-N.: Monotonicity, convexity, and Maclaurin series expansion of Qi’s normalized remainder of Maclaurin series expansion with relation to cosine, Open Math. 22(1) (2024), Art. 20240095.10.1515/math-2024-0095Search in Google Scholar

[18] Qi, F.: Absolute monotonicity of normalized tail of power series expansion of exponential function, Mathematics 12(18) (2024), Art. 2859.10.3390/math12182859Search in Google Scholar

[19] Qi, F.: Series and connections among central factorial numbers, Stirling numbers, inverse of Vandermonde matrix, and normalized remainders of Maclaurin series expansions, Mathematics 13(2) (2025), Art. 223.10.3390/math13020223Search in Google Scholar

[20] Qi, F.—Agarwal, R. P.—Lim, D.: Decreasing property of ratio of two logarithmic expressions involving tangent function, Math. Comput. Model. Dyn. Syst. 31(1) (2025), 1–16.10.1080/13873954.2024.2449322Search in Google Scholar

[21] Shuang, Y.—Guo, B.-N.—Qi, F.: Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115(3) (2021), Paper No. 135.10.1007/s13398-021-01071-xSearch in Google Scholar

[22] Temme, N. M.: Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996.10.1002/9781118032572Search in Google Scholar

[23] Wan, A.—Qi, F.: Power series expansion, decreasing property, and concavity related to logarithm of normalized tail of power series expansion of cosine, Electron. Res. Arch. 32(5) (2024), 3130–3144.10.3934/era.2024143Search in Google Scholar

[24] Wang, K. C.: The logarithmic concavity of (1 − 2^1−rζ(r), J. Changsha Comm. Univ. 14(2) (1998), 1–5 (in Chinese).Search in Google Scholar

[25] Wang, F.—Qi, F.: Power series expansion and decreasing property related to normalized remainders of power series expansion of sine, Filomat 38(29) (2024), 10447–10462.10.2298/FIL2429447WSearch in Google Scholar

[26] Zhang, H.-C.—Guo, B.-N.—Du, W.-S.: On Qi’s normalized remainder of Maclaurin power series expansion of logarithm of secant function, Axioms 13(12)(2024), Art. 860.10.3390/axioms13120860Search in Google Scholar

[27] Zhang, G.-Z.—Qi, F.: On convexity and power series expansion for logarithm of normalized tail of power series expansion for square of tangent, J. Math. Inequal. 18(3) (2024), 937–952.10.7153/jmi-2024-18-51Search in Google Scholar

[28] Zhang, G.-Z.—Yang, Z.-H.—Qi, F.: On normalized tails of series expansion of generating function of Bernoulli numbers, Proc. Amer. Math. Soc. 153(1) (2025), 131–141.10.1090/proc/16877Search in Google Scholar

[29] Zhang, T.—Yang, Z.-H.—Qi, F.—Du, W.-S.: Some properties of normalized tails of Maclaurin power series expansions of sine and cosine, Fractal Fract. 8(5) (2024), Art. 257.10.3390/fractalfract8050257Search in Google Scholar