Skip to main content
SpringerLink
Log in

Monotonicity and convexity involving generalized elliptic integral of the first kind

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind \({\mathcal {K}}_a(r)\) and its approximation \(\log [1+2/(ar')]\), and also the convexity (concavity) of their difference for \(a\in (0,1/2]\). As an application, we give new bounds for generalized Grötzsch ring function \(\mu _a(r)\) and a upper bound for \({\mathcal {K}}_a(r)\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Availability of supporting data

Not applicable.

References

  1. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  2. Alzer, H.: Some beta-function inequalities. Proc. R. Soc. Edinburgh. 133A(4), 731–745 (2003)

    Article  MathSciNet  Google Scholar 

  3. Alzer, H.: Inequalities for the beta function. Anal. Math. 40(1), 1–11 (2014)

    Article  MathSciNet  Google Scholar 

  4. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington (1964)

    MATH  Google Scholar 

  5. Anderson, G.D., Qiu, S.-L., Vamanamurthy, M.K., Vuorinen, M.: Generalized elliptic integrals and modular equations. Pac. J. Math. 192(1), 1–37 (2000)

    Article  MathSciNet  Google Scholar 

  6. Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23, 512–524 (1992)

    Article  MathSciNet  Google Scholar 

  7. Anderson, G.D., Qiu, S.-L., Vamanamurthy, M.K., Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)

    MATH  Google Scholar 

  8. Baricz, Á.: Turán type inequalities for generalized complete elliptic integrals. Math. Z. 256(4), 895–911 (2007)

    Article  MathSciNet  Google Scholar 

  9. Borwein, J.M., Borwein, P.B.: Pi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts. A Study in Analytic Number Theory and Computational Complexity; A Wiley-Interscience Publication. Wiley, New York (1987)

    Google Scholar 

  10. Chen, C.-P.: Inequalities and asymptotic expansions for the psi function and the Euler–Mascheroni constant. J. Number Theory. 163, 596–607 (2016)

    Article  MathSciNet  Google Scholar 

  11. Chen, C.-P.: Sharp inequalities and asymptotic series of a product related to the Euler–Mascheroni constant. J. Number Theory. 165, 314–323 (2016)

    Article  MathSciNet  Google Scholar 

  12. Guo, B.-N., Qi, F.: Monotonicity of functions connected with the gamma function and the volume of the unit ball. Integral Transforms Spec. Funct. 23(9), 701–708 (2012)

    Article  MathSciNet  Google Scholar 

  13. Huang, T.-R., Han, B.-W., Ma, X.-Y., Chu, Y.-M.: Optimal bounds for the generalized Euler–Mascheroni constant. J. Inequal. Appl. 2018, 9 (2018). (Article 118)

    Article  MathSciNet  Google Scholar 

  14. Huang, T.-R., Qiu, S.-L., Ma, X.-Y.: Monotonicity properties and inequalities for the generalized elliptic integral of the first kind. J. Math. Anal. Appl. 469(1), 95–116 (2019)

    Article  MathSciNet  Google Scholar 

  15. Hai, G.-J., Zhao, T.-H.: Monotonicity properties and bounds involving the two-parameter generalized Grötzsch ring function. J. Inequal. Appl. 2020, 17 (2020). (Article 66)

    Article  Google Scholar 

  16. Olverm, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  17. Qi, F.: Bounds for the ratio of two gamma functions. J. Inequal. Appl. Article ID 493058, 84 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Qi, F., Li, W.-H.: A logarithmically completely monotonic function involving the ratio of gamma functions. J. Appl. Anal. Comput. 5(4), 626–634 (2015)

    MathSciNet  MATH  Google Scholar 

  19. Qiu, S.-L., Vamanamurthy, M.K., Vuorinen, M.: Some inequalities for the growth of elliptic integrals. SIAM J. Math. Anal. 29, 1224–1237 (1998)

    Article  MathSciNet  Google Scholar 

  20. Richards, K.C.: A note on inequalities for the ratio of zero-balanced hypergeometric functions. Proc. Am. Math. Soc. Ser. B 6, 15–20 (2019)

    Article  MathSciNet  Google Scholar 

  21. Wang, M.-K., Chu, Y.-M., Qiu, S.-L.: Sharp bounds for generalized elliptic integrals of the first kind. J. Math. Anal. Appl. 429, 744–757 (2015)

    Article  MathSciNet  Google Scholar 

  22. Qian, W.-M., He, Z.-H., Chu, Y.-M.: Approximation for the complete elliptic integral of the first kind. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114(2), 57 (2020). https://doi.org/10.1007/s13398-020-00784-9

    Article  MathSciNet  MATH  Google Scholar 

  23. Yang, Z.-H.: A new way to prove L’Hospital monotone rules with applications. arXiv:1409.6408 [math.CA]

  24. Yang, Z.-H., Chu, Y.-M., Wang, M.-K.: Monotonicity criterion for the quotient of power series with applications. J. Math. Anal. Appl. 428(1), 587–604 (2015)

    Article  MathSciNet  Google Scholar 

  25. Yin, L., Huang, L.-G., Wang, Y.-L., Lin, X.-L.: An inequality for generalized complete elliptic integral. J. Inequal. Appl. 2017, 6 (2017). ( Article 303)

    Article  MathSciNet  Google Scholar 

  26. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind. J. Math. Anal. Appl. 462(2), 1714–1726 (2018)

    Article  MathSciNet  Google Scholar 

  27. Yang, Z.-H., Tian, J.-F.: Sharp inequalities for the generalized elliptic integrals of the first kind. Ramanujan J. 48(1), 91–116 (2019)

    Article  MathSciNet  Google Scholar 

  28. Yang, Z.-H., Tian, J.-F.: Convexity and monotonicity for elliptic integrals of the first kind and applications. Appl. Anal. Discret. Math. 13(1), 240–260 (2019)

    Article  MathSciNet  Google Scholar 

  29. Yang, Z.-H., Tian, J.-F.: Monotonicity, convexity, and complete monotonicity of two functions related to the gamma function. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113(4), 3603–3617 (2019)

    Article  MathSciNet  Google Scholar 

  30. Yang, Z.-H., Tian, J.-F., Wang, M.-K.: A positive answer to Bhatia–Li conjecture on the monotonicity for a new mean in its parameter. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114(3), 126 (2020). https://doi.org/10.1007/s13398-020-00856-w

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhao, T.-H., Wang, M.-K., Chu, Y.-M.: A sharp double inequality involving generalized complete elliptic integral of the first kind. AIMS Math. 5(5), 4512–4528 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202) and the Natural Science Foundation of Zhejiang Province (Grant No. LY19A010012).

Author information

Authors and Affiliations

  1. Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China

    Tie-Hong Zhao

  2. Department of Mathematics, Huzhou University, Huzhou, 313000, China

    Miao-Kun Wang & Yu-Ming Chu

Authors
  1. Tie-Hong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miao-Kun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu-Ming Chu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yu-Ming Chu.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, TH., Wang, MK. & Chu, YM. Monotonicity and convexity involving generalized elliptic integral of the first kind. RACSAM 115, 46 (2021). https://doi.org/10.1007/s13398-020-00992-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-020-00992-3

Keywords

Mathematics Subject Classification

Search

Navigation