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On the approximation of some special functions in Ramanujan’s generalized modular equation with signature 3\(^{*}\)

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Abstract

We study several special functions in Ramanujan’s generalized modular equation with signature 3. Some sharp inequalities for these functions, including the estimates for the solution of Ramanujan’s generalized modular equation with signature 3 and triplication inequality for the generalized Grötzsch ring function with two parameters, are derived.

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References

  1. Alzer, H., Richards, K.: On the modulus of the Grötzsch ring. J. Math. Anal. Appl. 432(1), 134–141 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

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

    MATH  Google Scholar 

  4. Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Topics in special functions. In: Papers on Analysis: A Volume Dedicated to Olli Martio on the Occasion of his 60th Birthday. Report, vol 83, pp. 5–26. Univ. Jyväskylä (2001)

  5. Berndt, B.C.: Ramanujan’s Notebooks II. Springer, New York (1989)

    Book  MATH  Google Scholar 

  6. Berndt, B.C., Bhargava, S., Garvan, F.G.: Ramanujan’s theories of elliptic functions to alternative bases. Trans. Am. Math. Soc. 347(11), 4163–4244 (1995)

    MathSciNet  MATH  Google Scholar 

  7. Borwein, J.M., Borwein, P.B.: Pi and AGM. Wiley, New York (1987)

    MATH  Google Scholar 

  8. Borwein, J.M., Borwein, P.B.: A remarkable cubic mean iteration. In: Computational Methods and Function Theory (Valparaíso 1989), Lecture Notes in Mathematics, vol 1435, pp 27–31. Springer, Berlin (1990)

  9. Borwein, J.M., Borwein, P.B.: A cubic counterpart of Jacobi’s identity and the AGM. Trans. Am. Math. Soc. 323(2), 691–701 (1991)

    MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chu, Y.-M., Wang, M.-K.: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3–4), 223–229 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chu, Y.-M., Wang, M.-K., Jiang, Y.-P., Qiu, S.-L.: Monotonicity, convexity, and inequalities involving the Agard distortion. Abstr. Appl. Anal. 2011, Article ID 671765 (2011)

  13. Chu, Y.-M., Wang, M.-K., Jiang, Y.-P., Qiu, S.-L.: Concavity of the complete elliptic integrals of the second kind with respect to Hölder means. J. Math. Anal. Appl. 395(2), 637–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chu, Y.-M., Wang, M.-K., Qiu, Y.-F.: On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function. Abstr. Appl. Anal. 2011, Article ID 697547 (2011)

  15. 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cooper, S., Ye, D.-X.: Level 14 and 15 analogues of Ramanujan’s elliptic functions to alternative bases. Trans. Am. Math. Soc. 368(11), 7883–7910 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Heikkala, V., Vamanamurthy, M.K., Vuorinen, M.: Generalized elliptic integrals. Comput. Methods Funct. Theory 9(1), 75–109 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huang, T.-R., Han, B.-W., Ma, X.-Y., Chu, Y.-M.: Optimal bounds for the generalized Euler–Mascheroni constant. J. Inequal. Appl. 2018, Paper No. 118 (2018)

  20. Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane, 2nd edn. Springer, New York (1973)

    Book  MATH  Google Scholar 

  21. Ma, X.-Y., Wang, M.-K., Zhong, G.-H., Qiu, S.-L., Chu, Y.-M.: Some inequalities for the generalized distortion functions. Math. Inequal. Appl. 15(4), 941–954 (2012)

    MathSciNet  MATH  Google Scholar 

  22. 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, Paper No. 274 (2017)

  23. Qiu, S.-L.: Grötzsch ring and Ramanujan’s modular equations. Acta Math. Sin. 43(2), 283–290 (2000). in Chinese

    MATH  Google Scholar 

  24. Qiu, S.-L., Qiu, Y.-F., Wang, M.-K., Chu, Y.-M.: Hölder mean inequalities for the generalized Grötzsch ring and Hersch–Pfluger distortion functions. Math. Inequal. Appl. 15(1), 237–245 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Qiu, S.-L., Vuorinen, M.: Infinite products and the normalized quotients of hypergeometric functions. SIAM J. Math. Anal. 30(5), 1057–1075 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ramanujan, S.: Collected Papers. Chelsea, New York (1962)

    Google Scholar 

  27. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)

    MATH  Google Scholar 

  28. Schultz, D.: Cubic theta functions. Adv. Math. 248, 618–697 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schultz, D.: Cubic modular equations in two variables. Adv. Math. 290, 329–363 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shen, L.-C.: A note on Ramanujan’s identities involving the hypergeometric function \(_{2}F_{1}(1/6,5/6;1;z)\). Ramanujan J. 30(2), 211–222 (2013)

    Article  MathSciNet  Google Scholar 

  31. Shen, L.-C.: On a theory of elliptic functions based on the incomplete integral of the hypergeometric function \(_{2}F_{1}(1/4,3/4;1/2;z)\). Ramanujan J. 34(2), 209–225 (2014)

    Article  MathSciNet  Google Scholar 

  32. Vuorinen, M.: Singular values, Ramanujan modular equations, and Landen transformations. Studia Math. 121(3), 221–230 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, M.-K., Chu, Y.-M.: Asymptotical bounds for complete elliptic integrals of the second kind. J. Math. Anal. Appl. 402(1), 119–126 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, M.-K., Chu, Y.-M.: Refinements of transformation inequalities for zero-balanced hypergeometric function. Acta Math. Sci. 37B(3), 607–622 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, M.-K., Chu, Y.-M.: Landen inequalities for a class of hypergeometric functions with applications. Math. Inequal. Appl. 21(2), 521–537 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Wang, M.-K., Chu, Y.-M., Jiang, Y.-P.: Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mt. J. Math. 46(2), 679–691 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, M.-K., Chu, Y.-M., Qiu, S.-L., Jiang, Y.-P.: Convexity of the complete elliptic integrals of the first kind with respect to Hölder means. J. Math. Anal. Appl. 388(2), 1141–1146 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, M.-K., Chu, Y.-M., Song, Y.-Q.: Ramanujan’s cubic transformation and generalized modular equation. Sci. China Math. 58(11), 2387–2404 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, M.-K., Chu, Y.-M., Song, Y.-Q.: Asymptotical formulas for Gaussian and generalized hypergeometric functions. Appl. Math. Comput. 276, 44–60 (2016)

    MathSciNet  MATH  Google Scholar 

  41. 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)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, M.-K., Qiu, S.-L., Chu, Y.-M.: Infinite series formula for Hübner upper bound function with applications to Hersch–Pfluger distortion function. Math. Inequal. Appl. 21(3), 629–648 (2018)

    MathSciNet  MATH  Google Scholar 

  43. Wang, M.-K., Qiu, S.-L., Jiang, Y.-P., Chu, Y.-M.: Generalized Hersch–Pfluger distortion function and complete elliptic integrals. J. Math. Anal. Appl. 385(1), 221–229 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: Inequalities for the generalized elliptic integrals and modular functions. J. Math. Anal. Appl. 331(2), 1275–1283 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: A power mean inequality for the Grötzsch ring function. Math. Inequal. Appl. 14(4), 833–837 (2011)

    MathSciNet  MATH  Google Scholar 

  46. Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: A power mean inequality involving the complete elliptic integrals. Rocky Mt. J. Math. 44(5), 1661–1667 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. Yang, Z.-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 

  48. 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  MATH  Google Scholar 

  49. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, Paper No. 210 (2017)

  50. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the error function. Math. Inequal. Appl. 21(2), 469–479 (2018)

    MathSciNet  MATH  Google Scholar 

  51. 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  MATH  Google Scholar 

  52. Yang, Z.-H., Zhang, W., Chu, Y.-M.: Sharp Gautschi inequality for parameter \(0<p<1\) with applications. Math. Inequal. Appl. 20(4), 1107–1120 (2017)

    MathSciNet  MATH  Google Scholar 

  53. Zhang, X.-H.: On the generalized modulus. Ramanujan J. 43(2), 405–413 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  54. Zhang, X.-M., Chu, Y.-M.: A double inequality for gamma function. J. Inequal. Appl. 2009, Article ID 503782 (2009)

  55. Zhang, X.-H., Wang, G.-D., Chu, Y.-M.: Some inequalities for the generalized Grötzsch function. Proc. Edinb. Math. Soc. (2) 51(1), 265–272 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. Zhang, X.-H., Wang, G.-D., Chu, Y.-M.: Remarks on generalized elliptic integrals. Proc. R. Soc. Edinb. 139A(2), 417–426 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  57. Zhang, X.-H., Wang, G.-D., Chu, Y.-M.: Convexity with respect to Hölder mean involving zero-balanced hypergeometric functions. J. Math. Anal. Appl. 353(1), 256–259 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  58. Zhao,T.-H., Chu, Y.-M.: A class of logarithmically completely monotonic functions associated with a gamma function. J. Inequal. Appl. 2010, Article ID 392431 (2010)

  59. Zhao, T.-H., Chu, Y.-M., Jiang, Y.-P.: Monotonic and logarithmically convex properties of a function involving gamma functions. J. Inequal. Appl. 2009, Article ID 728612 (2009)

  60. Zhao, T.-H., Chu, Y.-M., Wang, H.: Logarithmically complete monotonicity properties relating to the gamma function. Abstr. Appl. Anal. 2011, Article ID 896483 (2011)

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Authors and Affiliations

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

    Miao-Kun Wang

  2. College of Civil Engineering, Hunan University, Changsha, 410082, China

    Hong-Hu Chu

  3. College of Science, Hunan City University, Yiyang, 413000, China

    Yu-Ming Chu

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  1. Miao-Kun Wang
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  2. Hong-Hu Chu
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  3. Yu-Ming Chu
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Correspondence to Yu-Ming Chu.

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\(^{*}\)This work was supported by the Natural Science Foundation of China (Grant Nos. 11701176, 61373169, 11901061), and the Natural Science Foundation of Zhejiang Province (Grant No. LY19A010012).

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Wang, MK., Chu, HH. & Chu, YM. On the approximation of some special functions in Ramanujan’s generalized modular equation with signature 3\(^{*}\). Ramanujan J 56, 1–22 (2021). https://doi.org/10.1007/s11139-021-00437-4

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