Home Mathematics Several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals
Article
Licensed
Unlicensed Requires Authentication

Several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals

  • Wen-Hui Li and Bai-Ni Guo EMAIL logo
Published/Copyright: December 6, 2024
Become an author with De Gruyter Brill

Abstract

In the paper, in view of two monotonicity rules for the ratios of two functions and of two Maclaurin power series expansions, the authors establish several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals.


This paper is dedicated to Professor Dr. Feng Qi for his retirement in 2025


Acknowledgement

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

References

[1] Anderson, G. D.—Vamanamurthy, M. K.—Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.Search in Google Scholar

[2] Bagul, Y. J.—Dhaigude, R. M.—Chesneau, C.—Kostić, M.: Tight exponential bounds for hyperbolic tangent, Jordan J. Math. Stat. 15(4A) (2022), 807–821.Search in Google Scholar

[3] Banjac, B.—Malešević, B.—Mićović, M.—Mihailović, B.—Savatović, M.: The best possible constants approach for Wilker–Cusa–Huygens inequalities via stratification, Appl. Anal. Discrete Math. 18(1) (2024), 244–288;10.2298/AADM240308012BSearch in Google Scholar

[4] Biernacki, M.—Krzyż, J.: On the monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 9 (1955), 135–145.Search in Google Scholar

[5] Callan, D.: Solution to E 3306, Amer. Math. Monthly 98(10) (1991), 962–964.10.2307/2324162Search in Google Scholar

[6] Chen, S.—Liu, Z.: Automated proof of mixed trigonometric-polynomial inequalities, J. Symbolic Comput. 101 (2020), 318–329.10.1016/j.jsc.2019.10.002Search in Google Scholar

[7] Gearhart, W. B.—Shultz, H. S.: The function sinxx , College Math. J. 21(2) (1990), 90–99.10.2307/2686748Search in Google Scholar

[8] Guo, B.-N.—Li, W.—Qi, F.: Proofs of Wilker’s inequalities involving trigonometric functions. In: Inequality Theory and Applications, Vol. 3 (Chinju/Masan, 2001), Nova Science Publishers, Hauppauge, NY, 2003, 109–112.Search in Google Scholar

[9] Guo, B.-N.—Qiao, B.-M.—Qi, F.—Li, W.: On new proofs of Wilker’s inequalities involving trigonometric functions, Math. Inequal. Appl. 6(1) (2003), 19–22.10.7153/mia-06-02Search in Google Scholar

[10] Jiang, W.-D.—Luo, Q.-M.—And QI, F.: Refinements and sharpening of some Huygens and Wilker type inequalities, Turkish J. Anal. Number Theory 2 (2014), 134–139.10.12691/tjant-2-4-6Search in Google Scholar

[11] Li, W.-H.—Guo, B.-N.: Several inequalities for bounding sums of two (hyperbolic) sine cardinal functions, Filomat 38(11) (2024), 3937–3943.Search in Google Scholar

[12] Li, W.-H.—Miao, P.—Guo, B.-N.: Bounds for the NeumanSándor mean in terms of the arithmetic and contra-harmonic means, Axioms11(5) (2022), Art. No. 236.10.3390/axioms11050236Search in Google Scholar

[13] Li, W.-H.—Qi, F.: Harmonic mean inequalities for generalized hyperbolic functions, Montes Taurus J. Pure Appl. Math. 6(3) (2024), 199–207.Search in Google Scholar

[14] Li, W.-H.—Shen, Q.-X.—Guo, B.-N.: Several double inequalities for integer powers of the sinc and sinhc functions with applications to the Neuman–Sándor mean and the first Seiffert mean, Axioms 11(7) (2022), Art. No. 304.10.3390/axioms11070304Search in Google Scholar

[15] Malešević, B.—Makragić, M.: A method for proving some inequalities on mixed trigonometric polynomial functions, J. Math. Inequal. 10(3) (2016), 849–876.10.7153/jmi-10-69Search in Google Scholar

[16] Pinelis, I.: l’Hospital rules for monotonicity and the Wilker-Anglesio Inequality, Amer. Math. Monthly 111 (2004), 905–909.10.1080/00029890.2004.11920156Search in Google Scholar

[17] Sánchez-Reyes, J.: The hyperbolic sine cardinal and the catenary, College Math. J. 43(4) (2012), 285–290.10.4169/college.math.j.43.4.285Search in Google Scholar

[18] Sumner, J. S.—Jagers, A. A.—Vowe, M.—Anglesio, J.: Inequalities involving trigonometric functions, Amer. Math. Monthly 98(3) (1991), 264–267.10.2307/2325035Search in Google Scholar

[19] Wilker, J. B.: Problem E3306, Amer. Math. Monthly 96(1) (1989), 55–55.10.2307/2323260Search in Google Scholar

[20] Wu, S.—Debnath, L.: Wilker-type inequalities for hyperbolic functions, Appl. Math. Lett. 25(5) (2012), 837–842.10.1016/j.aml.2011.10.028Search in Google Scholar

[21] Wu, S.-H.—Srivastava, H. M.: A weighted and exponential generalization of Wilkers inequality and its applications, Integral Transforms Spec. Funct. 18(8) (2007), 529–535.10.1080/10652460701284164Search in Google Scholar

[22] Zhang, L.—Zhu, L.: A new elementary proof of Wilkers inequalities, Math. Inequal. Appl. 11(1) (2008), 149–151.10.7153/mia-11-09Search in Google Scholar

[23] Zhu, L.: New inequalities of Wilkers type for circular functions, AIMS Math. 5(5) (2020), 4874–4888.10.3934/math.2020311Search in Google Scholar

[24] Zhu, L.: New inequalities of Wilkers type for hyperbolic functions, AIMS Math. 5(1) (2020), 376–384.10.3934/math.2020025Search in Google Scholar

[25] Zhu, L.: On Wilker-type inequalities, Math. Inequal. Appl. 10(4) (2007), 727–731.10.7153/mia-10-67Search in Google Scholar

Received: 2024-02-08
Accepted: 2024-07-22
Published Online: 2024-12-06
Published in Print: 2024-12-15

© 2024 Mathematical Institute Slovak Academy of Sciences

Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2024-0104/html
Scroll to top button