Startseite Mathematik Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

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

  • Xin-Le Liu und Feng Qi EMAIL logo
Veröffentlicht/Copyright: 9. Juni 2025
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

In the paper, in view of the monotonicity rule for the ratio between two Maclaurin power series and by virtue of establishment of a monotonicity result for a sequence involving the ratio between two Bernoulli numbers, the authors investigate the monotonicity of the ratio between two normalized remainders of the Maclaurin power series expansion for the square of the tangent function.

2020 Mathematics Subject Classification: Primary 41A80; Secondary 26A48; 26D05; 33B10; 41A58

The second and corresponding author was partially supported by the Youth Project of Hulunbuir City for Basic Research and Applied Basic Research (Grant No. GH2024020).


Acknowledgement

The authors appreciate the anonymous referees for their careful reading, valuable comments, and helpful suggestions to the original version of this paper.

  1. (Communicated by Tomasz Natkaniec)

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.2015.05.006Suche 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.2014.12.009Suche 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/sym16080989Suche 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, 2004.10.1142/5605Suche 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, 2018.10.1515/9781400888252Suche 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.Suche 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), 797–804.10.1080/10652460902867718Suche 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.Suche in Google Scholar

[9] Guo, B.-N.—Qi, F.: The function (bx − ax/x: Logarithmic convexity and applications to extended mean values, Filomat 25(4) (2011), 63–73.10.2298/FIL1104063GSuche in Google Scholar

[10] Li, W.-H.—Lim, D.—Qi, F.: Expanding the function ln(1+ex 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.16.1.320-327Suche 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-2023-0159Suche 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/axioms13050317Suche 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/sym15091686Suche 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-2022-16-33Suche 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/math11143107Suche 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/math12121781Suche 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-2024-0095Suche 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/math12182859Suche 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/math13020223Suche 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), 1–16.10.1080/13873954.2024.2449322Suche 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-xSuche 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, 1996.10.1002/9781118032572Suche 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.2024143Suche in Google Scholar

[24] Wang, K. C.: The logarithmic concavity of (1 − 21−rζ(r), J. Changsha Comm. Univ. 14(2) (1998), 1–5 (in Chinese).Suche 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/FIL2429447WSuche 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/axioms13120860Suche 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-2024-18-51Suche 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/16877Suche 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/fractalfract8050257Suche in Google Scholar

Received: 2024-10-06
Accepted: 2025-01-21
Published Online: 2025-06-09
Published in Print: 2025-06-26

© 2025 Mathematical Institute Slovak Academy of Sciences

Heruntergeladen am 15.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2025-0051/html
Button zum nach oben scrollen