• P-ISSN 0974-6846 E-ISSN 0974-5645

Indian Journal of Science and Technology


Indian Journal of Science and Technology

Year: 2024, Volume: 17, Issue: 9, Pages: 787-793

Original Article

On Wilker’s and Huygen’s Type Inequalities for Generalized Trigonometric and Hyperbolic Functions

Received Date:22 December 2023, Accepted Date:27 January 2024, Published Date:20 February 2024


Objectives: The Trigonometric inequalities, generalized trigonometric inequalities which have been obtained by Wilker and Cusa Huygens have attracted attention of so many researchers. Generalized trigonometric functions are simple generalization of the classical trigonometric functions. It is related to the r- Laplacian, which is known as a non-linear differential operator. Method: For the establishment of inequalities involving generalized trigonometric and hyperbolic functions convexity plays the important role in many aspects, also Monotonicity rule is used for sharpness of inequalities. This technique is used to refine and sharpness of inequalities. Findings: Our main result of this paper focus on generalization of Wilker and Cusa Huygens type inequalities for generalized trigonometric and hyperbolic functions with one parameter. Novelty: The inequalities with generalized trigonometric and hyperbolic functions proved in this research paper are Wilker's and Cusa Huygens generalization. It can be used for further refinement and sharpness.

Keywords: Trigonometric function, Hyperbolic function, Generalized Trigonometric, Hyperbolic functions, Wilker Inequality and Huygen's Inequality


  1. Nantomah K. Cusa-Huygens, Wilker and Huygens Type Inequalities for Generalized Hyperbolic Functions. Earthline Journal of Mathematical Sciences. 2021;5(2):277–289. Available from: https://doi.org/10.34198/ejms.5221.277289
  2. Bagul YJ, Dhaigude RM, Bhayo BA, Raut VM. Wilker and Huygens type inequalities for mixed trigonometric-hyperbolic functions. Tbilisi Mathematical Journal . 2021;14(2):207–220. Available from: https://doi.org/10.32513/tmj/19322008134
  3. Wang MK, Hong MY, Xu YF, Shen ZH, Chu YM. Inequalities for Generalized Trigonometric and Hyperbolic Functions with One Parameter. Journal of Mathematical Inequalities. 2020;14(1):1–21. Available from: https://files.ele-math.com/articles/jmi-14-01.pdf
  4. Ma X, Si X, Zhong G, He J. Inequalities for the generalized trigonometric and hyperbolic functions. Open Mathematics. 2020;18(1):1580–1589. Available from: https://doi.org/10.1515/math-2020-0096
  5. Yin L, Huang L, Lin X. Inequalities and bounds for the p-generalized trigonometric functions. AIMS Mathematics. 2021;6(10):11097–11108. Available from: https://doi.org/10.3934/math.2021644
  6. Takeuchi S. Applications of generalized trigonometric functions with two parameters II. Differential Equations & Applications. 2019;11(4):563–575. Available from: https://files.ele-math.com/articles/dea-11-28.pdf
  7. Zhu L. New Inequalities of Cusa–Huygens Type. Mathematics. 2021;9(17):1–13. Available from: https://doi.org/10.3390/math9172101
  8. Shinde R, Chesneau C, Darkunde N, Ghodechor S, Lagad A. Revisit of an Improved Wilker Type Inequality. Pan-American Journal of Mathematics. 2023;2:1–17. Available from: https://doi.org/10.28919/cpr-pajm/2-13
  9. Darkunde N, Ghodechor S. Inequalities Involving Generalized Trigonometric And Hyperbolic. jnanabha. 2023;53(02):301–306. Available from: https://doi.org/10.58250/jnanabha.2023.53236


© 2024 Darkunde & Ghodechor. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Published By Indian Society for Education and Environment (iSee)


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