Sharp Approximations for the Ramanujan Constant

  • Song-Liang QiuEmail author
  • Xiao-Yan Ma
  • Ti-Ren Huang


In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan R-function) R(a), by showing some monotonicity, concavity and convexity properties of certain combinations defined in terms of R(a), \(\sin (\pi a)\) and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too.


The Ramanujan constant Monotonicity Convexity and concavity Approximation Functional inequalities The Riemann zeta function 

Mathematics Subject Classification

11M06 33B15 33C05 33F05 



  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1965)zbMATHGoogle Scholar
  2. 2.
    Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Mappings. Wiley, New York (1997)zbMATHGoogle Scholar
  3. 3.
    Anderson, G.D., Barnard, R.W., Richards, K.C., et al.: Inequalities for zero-balanced hypergeometric functions. Trans. Am. Math. Soc. 347, 1713–1723 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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
  5. 5.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and Its Applications. Cambridge Univ. Press, Cambridge (1999)Google Scholar
  6. 6.
    Balasubramanian, R., Ponnusamy, S., Vuorinen, M.: Functional inequalities for quotients of hypergeometric functions. J. Math. Anal. Appl. 218, 256–268 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ponnusamy, S., Vuorinen, M.: Asymptotic expansions and inequalities for hypergeometric function. Mathematika 44, 278–301 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Qiu, S.L.: Singular values, quasiconformal maps and the Schottky upper bound. Sci. China (Ser. A) 41(12), 1241–1247 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Qiu, S.L.: Grötzsch ring and Ramanujan’s modular equations. Acta Math. Sin. 43(2), 283–290 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Qiu, S.-L., Feng, B.-P.: Some properties of the Ramanujan constant. J. Hangzhou Dianzi Univ. 27(3), 88–91 (2007)Google Scholar
  11. 11.
    Qiu, S.-L., Ma, X.-Y., Huang, T.-R.: Some properties of the difference between the Ramanujan constant and beta function. J. Math. Anal. Appl. 446, 114–129 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Qiu, S.L., Vuorinen, M.: Infinite products and the normalized quotients of hypergeometric function. SIAM J. Math. Anal. 30, 1057–1075 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Qiu, S.L., Vuorinen, M.: Handbook of Complex Analysis: Special Function in Geometric Function Theory, pp. 621–659. Elsevier, Amsterdam (2005)zbMATHGoogle Scholar
  14. 14.
    Wang, M.K., Chu, Y.M., Qiu, S.L.: Some monotonicity properties of generalized elliptic integrals with applications. Math. Inequal. Appl. 3, 671–677 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhou, P.G., Qiu, S.L., Tu, G.Y., Li, Y.L.: Some properties of the Ramanujan constant. J. Zhejiang Sci. Technol. Univ. 27(5), 835–841 (2010)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Sci-Tech UniversityHangzhouChina

Personalised recommendations