The Ramanujan Journal

, Volume 46, Issue 1, pp 189–200 | Cite as

Inequalities and infinite product formula for Ramanujan generalized modular equation function

  • Miao-Kun Wang
  • Yong-Min Li
  • Yu-Ming ChuEmail author


We present several inequalities for the Ramanujan generalized modular equation function \(\mu _{a}(r)=\pi F(a,1-a;1;1-r^2)/\) \([2\sin (\pi a)F(a,1-a;1;r^2)]\) with \(a\in (0,1/2]\) and \(r\in (0,1)\), and provide an infinite product formula for \(\mu _{1/4}(r)\), where \(F(a,b;c;x)={}_{2}F_{1}(a,b;c;x)\) is the Gaussian hypergeometric function.


Gaussian hypergeometric function Ramanujan generalized modular equation Quadratic transformation Infinite product 

Mathematics Subject Classification

33C05 11F03 


  1. 1.
    Alzer, H., Richards, K.: On the modulus of the Grötzsch ring. J. Math. Anal. Appl. 432(1), 134–141 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)zbMATHGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barnard, R.W., Pearce, K., Richards, K.C.: A monotonicity property involving \({}_{3}F_{2}\) and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32(2), 403–419 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berndt, B.C.: Ramanujan’s Notebooks. Part I. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  6. 6.
    Berndt, B.C.: Ramanujan’s Notebooks. Part II. Springer, New York (1989)zbMATHGoogle Scholar
  7. 7.
    Berndt, B.C.: Ramanujan’s Notebooks. Part III. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  8. 8.
    Berndt, B.C.: Ramanujan’s Notebooks. Part IV. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  9. 9.
    Berndt, B.C., Bhargave, S., Garvan, F.G.: Ramanujan’s theories of elliptic functions to alternative bases. Trans. Am. Math. Soc. 347(11), 4163–4244 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bhayo, B.A., Vuorinen, M.: On generalized complete elliptic integrals and modular functions. Proc. Edinb. Math. Soc. (2) 55(3), 591–611 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Borwein, J.M., Borwein, P.B.: Pi and AGM. Wiley, New York (1987)zbMATHGoogle Scholar
  12. 12.
    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)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Borwein, D., Borwein, J.M., Glasser, M.L., Wan, J.G.: Moments of Ramanujan’s generalized elliptic integrals and extensions of Catalan’s constant. J. Math. Anal. Appl. 384(2), 478–496 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chan, H.H., Liu, Z.G.: Analogues of Jacobi’s inversion formula for incomplete elliptic integrals of the first kind. Adv. Math. 174(1), 69–88 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  16. 16.
    Qiu, S.L., Vuorinen, M.: Infinite products and the normalized quotients of hypergeometric functions. SIAM J. Math. Anal. 30(5), 1057–1075 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Qiu, S.L., Vuorinen, M.: Duplication inequalities for the ratios of hypergeometric functions. Forum Math. 12(1), 109–133 (2000)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Saigo, M., Srivastava, H.M.: The behavior of the zero-balanced hypergeometric series \({}_{p}F_{p-1}\) near the boundary of its convergence region. Proc. Am. Math. Soc. 110(1), 71–76 (1990)zbMATHGoogle Scholar
  19. 19.
    Schultz, D.: Cubic theta functions. Adv. Math. 248, 618–697 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schultz, D.: Cubic modular equations in two variables. Adv. Math. 290, 329–363 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shen, L.C.: On an identity of Ramanujan based on the hypergeometric series \(_{2}F_{1}(1/3,2/3;1/2;x)\). J. Number Theory 69(2), 125–134 (1998)MathSciNetCrossRefGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Venkatachaliengar, K.: Development of Elliptic Functions According to Ramanujan. Technical Report 2. Madurai Kamaraj University, Madurai (1988)zbMATHGoogle Scholar
  24. 24.
    Vuorinen, M.: Singular values, Ramanujan modular equations, and Landen transformations. Stud. Math. 121(3), 221–230 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wang, G.D., Zhang, X.H., Jiang, Y.P.: Concavity with respect to Hölder means involving the generalized Grötzsch function. J. Math. Anal. Appl. 379(1), 200–204 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wang, M.K., Chu, Y.M., Jiang, Y.P., Yan, D.D.: A class of quadratic transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 34A(4), 999–1007 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, London (1962)zbMATHGoogle Scholar
  29. 29.
    Zhang, X.H.: Solution to a conjecture on the Legendre \(M\)-function with an applications to the generalized modulus. J. Math. Anal. Appl. 431(2), 1190–1196 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zhang, X.H.: On the generalized modulus. Ramanujan J. (2016). doi: 10.1007/s11139-015-9746-0 Google Scholar
  31. 31.
    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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsHuzhou UniversityHuzhouChina

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