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

Abstract

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.

References

  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)

  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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Anderson, G.D.; Qiu, S.-L.; Vamanamurthy, M.K.: Elliptic integrals inequalities, with applications. Constr. Approx. 14, 195–207 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Anderson, G.D.; Vamanamurthy, M.K.: Some properties of quasiconformal distortion functions. N. Z. J. Math. 24, 1–16 (1995)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Anderson, G.D.; Duren, P.; Vamanamurthy, M.K.: An inequality for elliptic integrals. J. Math. Anal. Appl. 182, 257–259 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Anderson, G.D.; Vamanamurthy, M.K.; Vuorinen, M.: Conformal Invariants, Inequalities and Quasiconformal Maps, p. 505. Wiley, New York (1997)

    Google Scholar 

  8. 8.

    Andrews, G.E.; Askey, R.; Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  9. 9.

    Baricz, Á.; Bhayo, B.A.; Klén, R.: Convexity properties of generalized trigonometric and hyperbolic functions. Aequat. Math. 89, 473–484 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Bhayo, B.A.; Vuorinen, M.: On generalized trigonometric functions with two parameters. J. Approx. Theory 164, 1415–1426 (2012)

    MathSciNet  MATH  Article  Google 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)

  13. 13.

    Brent, R.P.: Fast multiple-precision evaluation of elementary functions. J. Assoc. Comput. Math. 23(2), 242–251 (1976)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Bushell, P.J.; Edmunds, D.E.: Remarks on generalised trigonometric functions. Rocky Mt. J. Math. 42, 13–52 (2012)

    MATH  Article  Google Scholar 

  15. 15.

    Carlson, F.; Gustafson, J.L.: Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. Anal. 25, 288–303 (1994)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MATH  Google Scholar 

  21. 21.

    Edmunds, D.E.; Gurka, P.; Lang, J.: Properties of generalized trigonometric functions. J. Approx. Theory 164, 47–56 (2012)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I.: Integrals and Series. Gordon and Breach Science Publishers, Amsterdam (1990)

    Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Salamin, E.: Computation of \(\pi \) using arithmetic–geometric means. Math. Comput. 30(135), 565–570 (1976)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Takeuchi, S.: A new form of the generalized complete elliptic integrals. Kodai Math. J. 39(1), 202–226 (2016)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Google 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)

  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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Barkat Ali Bhayo.

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Bhayo, B.A., Yin, L. On a function involving generalized complete (pq)-elliptic integrals. Arab. J. Math. 9, 73–82 (2020). https://doi.org/10.1007/s40065-019-0242-z

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Mathematics Subject Classification

  • 33C99
  • 33B99