Research article

Inequalities and bounds for the $ p $-generalized trigonometric functions

  • Received: 24 April 2021 Accepted: 29 July 2021 Published: 03 August 2021
  • MSC : 33B10

  • In this paper, we mainly show some bounds and inequalities for the $ p $-generalized trigonometric functions defined by Richter.

    Citation: Li Yin, Liguo Huang, Xiuli Lin. Inequalities and bounds for the $ p $-generalized trigonometric functions[J]. AIMS Mathematics, 2021, 6(10): 11097-11108. doi: 10.3934/math.2021644

    Related Papers:

  • In this paper, we mainly show some bounds and inequalities for the $ p $-generalized trigonometric functions defined by Richter.



    加载中


    [1] Á. Baricz, B. A. Bhayo, T. K. Pogány, Functional inequalities for generalized inverse trigonometric and hyperbolic functions, J. Math. Anal. Appl., 417 (2014), 244-259. Available from: http://arXiv.org/abs/1401.4863.
    [2] Á. Baricz, B. A. Bhayo, M. Vuorinen, Turán type inequalities for generalized inverse trigonometric functions, Filomat, 29 (2015), 303-313. Available from: http://arXiv.org/abs/1209.1696.
    [3] Á. Baricz, B. A. Bhayo, R. Klén, Convexity properties of generalized trigonometric and hyperbolic functions, Aequat. Math., 89 (2015), 473-484. Available from: http://arXiv.org/abs/1301.0699.
    [4] Á. Baricz, B. A. Bhayo, T. K. Pogány, Convexity properties of generalized trigonometric and hyperbolic functions, J. Math. Anal. Appl., 417 (2014), 244-259. doi: 10.1016/j.jmaa.2014.03.039
    [5] B. A. Bhayo, M. Vuorinen, Inequalities for eigenfunctions of the $p$-Laplacian, Issues of Analysis, 20 (2013), 13-35. Available from: http://arXiv.org/abs/1101.3911.
    [6] B. A. Bhayo, M. Vuorinen, Power mean inequalities generalized trigonometric functions, Math. Vesnik, 67 (2015), 17-25. Available from: http://arXiv.org/abs/1209.0983.
    [7] A. Elbert, A half-linear second order differential equation. Colloqia Mathematica Societatis Jonos Bolyai, 30. Qualitiative Theory of Differential Equations, Szeged (Hungary) (1979), 153-180.
    [8] M. R. Eslahchi, M. Masjed-Jamei, E. Babolian, On numerical improvement of Gauss-Lobatto quadrature rules, Appl. Math. Comput., 164 (2005), 707-717.
    [9] M. Dehghan, M. Masjed-Jamei, M. R. Eslahchi, On numerical improvement of closed Newton-Cotes quadrature rules, Appl. Math. Comput., 165 (2005), 251-260.
    [10] R. Klén, M. Vuorinen, X. H. Zhang, Inequalities for the generalized trigonometric and hyperbolic functions, J. Math. Anal. Appl., 409 (2014), 521-529. Available from: http://arXiv.org/abs/1210.6749.
    [11] B. A. Bhayo, L. Yin, Logarithmic mean inequality for generalized trigonometric and hyperbolic functions, Acta. Univ. Sapientiae Math., 6 (2014), 135-145. Available from: http://arXiv.org/abs/1404.6732.
    [12] B. A. Bhayo, L. Yin, On the generalized convexity and concavity, Probl. Anal., 22 (2015), 1-9. Available from: http://arXiv.org/abs/1411.6586.
    [13] B. A. Bhayo, L. Yin, On the conjecture of generalized trigonometric and hyperbolic functions, Math. Pannon., 24 (2013), 1-8. Available from: http://arXiv.org/abs/1402.7331.
    [14] M. Masjed-Jamei, S. S. Dragomir, A new generalization of the Ostrowski inequality and applications, Filomat, 25 (2011), 115-123. doi: 10.2298/FIL1101115M
    [15] M. Masjed-Jamei, S. S. Dragomir, H. M. Srivastava, Some generalizations of the Cauchy-Schwarz and the Cauchy-Bunyakovsky inequalities involving four free parameters and their applications, Math. Comput. Model., 49, (2009), 1960-1968.
    [16] L. G. Huang, L. Yin, Y. L. Wang, X. L. Lin, Some new Wilker and Cusa type inequalities for generalized trigonometric and hyperbolic functions, J. Inequal. Appl., 2018 (2018), 52. doi: 10.1186/s13660-018-1644-8
    [17] D. B. Karp, E. G. Prilepkina, Parameter convexity and concavity of generalized trigonometric functions, J. Math. Anal. Appl., 421 (2015), 370-382. Available from: http://arXiv.org/abs/1402.3357.
    [18] J. C. Kuang, Applied inequalities (Second edition). Shan Dong Science and Technology Press. Jinan, 2002.
    [19] P. Lindqvist, Some remarkable sine and cosine functions, Ricerche di Math., XLIV (1995), 269-290.
    [20] T. J. Mildorf, A sharp bound on the two variable powers mean, Math. Reflections, 2 (2006), 3-7.
    [21] D. S. Mitrinovic, Analytic inequalities, Springer-verlag, New York, 1970.
    [22] W. D. Richter, Generalized spherical and simplicial coordinates, J. Math. Anal. Appl., 336 (2007), 1187-1202. doi: 10.1016/j.jmaa.2007.03.047
    [23] W. D. Richter, Continuous $l_{n, p}$-symmetric distributions, Lithuanian Math. J., 49 (2009), 93-108.
    [24] J. Sándor, On some new Wilker and Huygens type trigonometric-hyperbolic inequalities, Proceedings of the Jangjeon Mathematical Society, 15 (2012). Available from: http://arXiv.org/abs/1105.0859.
    [25] S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with $p$-Laplacian, J. Math. Anal. Appl., 385 (2012), 24-35. doi: 10.1016/j.jmaa.2011.06.063
    [26] T. Kamiya, S. Takeuchi, Complete $(p, q)$-elliptic integrals with application to a family of means, J. Classical Anal., 10 (2015), 15-25. Available from: http://arXiv.org/abs/1507.01383.
    [27] S. Takeuchi, The complete $p$-elliptic integrals and a computation formula of $\pi_p$ for $p = 4$, Ramanujan J., 46 (2018), 309-321. Available from: http://arXiv.org/abs/1503.02394.
    [28] S. Takeuchi, A new form of the generalized complete elliptic integrals, Kodai J. Math., 39 (2016), 202-226. Available from: http://arXiv.org/abs/1411.4778.
    [29] S. Takeuchi, Multiple-angle formulas of generalized trigonometric functions with two parameters, J. Math. Anal. Appl., 444 (2016), 1000-1014. Available from: http://arXiv.org/abs/1603.06709.
    [30] S. Takeuchi, Legendre-type relations for generalized complete elliptic integrals, J. Classical Anal., 9 (2016), 35-42. Available from: http://arXiv.org/abs/1606.05115.
    [31] M. K. Wang, Z. Y. He, Y. M. Chu, Sharp Power Mean Inequalities for the Generalized Elliptic Integral of the First Kind, Comput. Methods Funct. Theory, 20 (2020), 111-124. doi: 10.1007/s40315-020-00298-w
    [32] M. K. Wang, H. H. Chu, Y. M. Chu, Precise bounds for the weighted Hölder mean of the complete $p$-elliptic integrals, J. Math. Anal. Appl., 480 (2019), 123388. doi: 10.1016/j.jmaa.2019.123388
    [33] M. K. Wang, W. Zhang, Y. M. Chu, Monotonicity, convexity and inequalities involving the generalized elliptic integrals, Acta Math. Sci., 39(B) (2019), 1440-1450.
    [34] T. R. Huang, S. Y. Tan, X. Y. Ma, Y. M. Chu, Monotonicity properties and bounds for the complete p-elliptic integrals, J. Inequal. Appl., 2018 (2018), Article ID 239, 11.
    [35] Z. H. Yang, Y. M. Chu, A monotonicity property involving the generalized elliptic integral of the first kind, Math. Inequal. Appl., 20 (2017), 729-735.
    [36] L. Yin, L. G. Huang, Y. L. Wang, X. L. Lin, A survey for generalized trigonometric and hyperbolic functions, J. Math. Inequal., 13 (2019), 833-854.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1543) PDF downloads(95) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog