On a function involving generalized complete (pq)-elliptic integrals

  • Barkat Ali BhayoEmail author
  • Li Yin
Open Access


Motivated by the work of Alzer and Richards (Anal Math 41:133–139, 2015), here authors study the monotonicity and convexity properties of the function
$$\begin{aligned} \Delta _{p,q} (r) = \frac{{E_{p,q}(r) - \left( {r'} \right) ^p K_{p,q}(r) }}{{r^p }} - \frac{{E'_{p,q}(r) - r^p K'_{p,q}(r) }}{{\left( {r'} \right) ^p }}, \end{aligned}$$
where \(K_{p,q}\) and \(E_{p,q}\) denote the complete (pq)-elliptic integrals of the first and second kind, respectively.

Mathematics Subject Classification

33C99 33B99 



  1. 1.
    Abramowitz, M.; Stegun, I. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards. Dover, New York (1965)Google Scholar
  2. 2.
    Alzer, H.; Qiu, S.-L.: Monotonicity theorems and inequalities for the generalized complete elliptic integrals. J. Comput. Appl. Math. 172(2), 289–312 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alzer, H.; Richards, K.: A note on a function involving complete elliptic integrals: monotonicity, convexity, inequalities. Anal. Math. 41, 133–139 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Anderson, G.D.; Qiu, S.-L.; Vamanamurthy, M.K.: Elliptic integrals inequalities, with applications. Constr. Approx. 14, 195–207 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Anderson, G.D.; Vamanamurthy, M.K.: Some properties of quasiconformal distortion functions. N. Z. J. Math. 24, 1–16 (1995)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Anderson, G.D.; Duren, P.; Vamanamurthy, M.K.: An inequality for elliptic integrals. J. Math. Anal. Appl. 182, 257–259 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Anderson, G.D.; Vamanamurthy, M.K.; Vuorinen, M.: Conformal Invariants, Inequalities and Quasiconformal Maps, p. 505. Wiley, New York (1997)zbMATHGoogle Scholar
  8. 8.
    Andrews, G.E.; Askey, R.; Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)zbMATHCrossRefGoogle Scholar
  9. 9.
    Baricz, Á.; Bhayo, B.A.; Klén, R.: Convexity properties of generalized trigonometric and hyperbolic functions. Aequat. Math. 89, 473–484 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bhayo, B.A.; Vuorinen, M.: On generalized trigonometric functions with two parameters. J. Approx. Theory 164, 1415–1426 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bhayo, B.A.; Yin, L.: On generalized \((p,q)\)-elliptic integrals. arXiv:1507.00031 [math.CA]
  12. 12.
    Borwein, J.M.; Borwein, P.B.: Pi and the AGM, A study in analytic number theory and computational complexity, Reprint of the 1987 original. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 4. A Wiley-Interscience Publication. Wiley, New York (1998)Google Scholar
  13. 13.
    Brent, R.P.: Fast multiple-precision evaluation of elementary functions. J. Assoc. Comput. Math. 23(2), 242–251 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bushell, P.J.; Edmunds, D.E.: Remarks on generalised trigonometric functions. Rocky Mt. J. Math. 42, 13–52 (2012)zbMATHCrossRefGoogle Scholar
  15. 15.
    Carlson, F.; Gustafson, J.L.: Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. Anal. 25, 288–303 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Chu, Y.-M.; Wang, M.-K.: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3–4), 223–229 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Chu, Y.-M.; Wang, M.-K.; Qiu, S.-L.: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41–51 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Chu, Y.-M.; Qiu, Y.-F.; Wang, M.-K.: Hölder mean inequalities for the complete elliptic integrals. Integral Transforms Spec. Funct. 23(7), 521–527 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Chu, Y.-M.; Wang, M.-K.; Qiu, S.-L.; Jiang, Y.-P.: Bounds for complete elliptic integrals of the second kind with applications. Comput. Math. Appl. 63(7), 1177–1184 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Drábek, P.; Manásevich, R.: On the closed solution to some \(p\)-Laplacian nonhomogeneous eigenvalue problems. Differ. Int. Equ. 12, 723–740 (1999)zbMATHGoogle Scholar
  21. 21.
    Edmunds, D.E.; Gurka, P.; Lang, J.: Properties of generalized trigonometric functions. J. Approx. Theory 164, 47–56 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Huang, T.-R.; Tan, S.-Y.; Zhang, X.-H.: Monotonicity, convexity, and inequalities for the generalized elliptic integrals. J. Inequal. Appl. 2017, Article ID 278 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Huang, T.-R.; Tan, S.-Y.; Ma, X.-Y.; Chu, Y.-M.: Monotonicity properties and bounds for the complete p-elliptic integrals. J. Inequal. Appl. 2018, Article ID 239 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kamiya, T.; Takeuchi, S.: Complete \((p, q)\)-elliptic integrals with application to a family of means. J. Class. Anal. 10(1), 15–25 (2017)MathSciNetGoogle Scholar
  25. 25.
    Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I.: Integrals and Series. Gordon and Breach Science Publishers, Amsterdam (1990)zbMATHGoogle Scholar
  26. 26.
    Qian, W.-M.; Chu, Y.-M.: Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters. J. Inequal. Appl. 2017, Article ID 274 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Salamin, E.: Computation of \(\pi \) using arithmetic–geometric means. Math. Comput. 30(135), 565–570 (1976)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Takeuchi, S.: Generalized Jacobian elliptic functions and their application to bifurcation problems associated with \(p\)-Laplacian. J. Math. Anal. Appl. 385, 24–35 (2012)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Takeuchi, S.: A new form of the generalized complete elliptic integrals. Kodai Math. J. 39(1), 202–226 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Takeuchi, S.: The complete \(p\)-elliptic integrals and a computation formula of \(\pi _p\) for \(p=4\). Ramanujan J. 46(2), 309–321 (2018)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Wang, M.-K.; Chu, Y.-M.: Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 37B(3), 607–622 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Wang, M.-K.; Chu, Y.-M.: Landen inequalities for a class of hypergeometric functions with applications. Math. Inequal. Appl. 21(2), 52–537 (2018)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Wang, Zh-X; Guo, D.-R.: Introduction to Special Function. Peking University Press, Beijing (2004)Google Scholar
  34. 34.
    Wang, G.-D.; Zhang, X.-H.; Chu, Y.-M.: Inequalities for the generalized elliptic integrals and modular equations. J. Math. Anal. Appl. 331(2), 1275–1283 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Wang, M.-K.; Chu, Y.-M.; Qiu, Y.-F.; Qiu, S.-L.: An optimal power mean inequality for the complete elliptic integrals. Appl. Math. Lett. 24(6), 887–890 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Wang, M.-K.; Qiu, S.-L.; Chu, Y.-M.: Infinite series formula for Hübner upper bound function with applications to Hersch–Puger distortion function. Math. Inequal. Appl. 21(3), 629–648 (2018)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Wang, M.-K.; Li, Y.-M.; Chu, Y.-M.: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. 46(1), 189–200 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Yang, Zh-H; Chu, Y.-M.: A monotonicity property involving the generalized elliptic integral of the first kind. Math. Inequal. Appl. 20(3), 729–735 (2017)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Yang, Zh.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: Monotonicity rule for the quotient of two functions and its application. J. Inequal. Appl., Article 106 (2017)Google Scholar
  40. 40.
    Yang, Zh-H; Qian, W.-M.; Chu, Y.-M.: Monotonicity properties and bounds involving the complete elliptic integrals of the first kind. Math. Inequal. Appl. 21(4), 1185–1199 (2018)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Yang, Zh-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)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of mathematicsSukkur IBA UniversitySindhPakistan
  2. 2.Faculty of Engineering and natural sciencesSabanci UniversityTuzlaTurkey
  3. 3.Department of MathematicsBinzhou UniversityBinzhouChina

Personalised recommendations