Advertisement

The Ramanujan Journal

, Volume 46, Issue 1, pp 173–188 | Cite as

On geometric properties of the generating function for the Ramanujan sequence

  • Andrew Bakan
  • Stephan Ruscheweyh
  • Luis Salinas
Article
  • 163 Downloads

Abstract

The Ramanujan sequence \(\{\theta _{n}\}_{n \ge 0}\), defined as \(\theta _{0}= {1}/{2}\), \({n^{n}} \theta _{n}/{n !} = {e^{n}}/{2} - \sum _{k=0}^{n-1} {n^{k}}/{k !}\, \), \(n \ge 1\), has been studied on many occasions and in many different contexts. Adell and Jodrá (Ramanujan J 16:1–5, 2008) and Koumandos (Ramanujan J 30:447–459, 2013) showed, respectively, that the sequences \(\{\theta _{n}\}_{n \ge 0}\) and \(\{4/135 - n \cdot (\theta _{n}- 1/3 )\}_{n \ge 0}\) are completely monotone. In the present paper, we establish that the sequence \(\{(n+1) (\theta _{n}- 1/3 )\}_{n \ge 0}\) is also completely monotone. Furthermore, we prove that the analytic function \((\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} / n^{\alpha }\) is universally starlike for every \(\alpha \ge 1\) in the slit domain \(\mathbb {C}\setminus [1,\infty )\). This seems to be the first result putting the Ramanujan sequence into the context of analytic univalent functions and is a step towards a previous stronger conjecture, proposed by Ruscheweyh et al. (Israel J Math 171:285–304, 2009), namely that the function \((\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} \) is universally convex.

Keywords

Ramanujan’s sequence Completely monotone sequences Universally starlike functions Universally convex functions 

Mathematics Subject Classification

Primary 33B10 Secondary 30C45 

References

  1. 1.
    Alm, S.E.: Monotonicity of the difference between median and mean of gamma distributions and a related Ramanujan sequence. Bernoulli 9, 351–371 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adell, J.A., Jodrá, P.: On the complete monotonicity of a Ramanujan sequence connected with \(e^{n}\). Ramanujan J. 16, 1–5 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alzer, H.: On Ramanujan’s inequalities for \(\exp (k)\). J. Lond. Math. Soc. 69, 639–656 (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bakan, A., Ruscheweyh, S., Salinas, L.: Universal convexity and universal starlikeness of polylogarithms. Proc. Am. Math. Soc. 143, 717–729 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Flajolet, P., Grabner, P.J., Kirschenhofer, P., Prodinger, H.: On Ramanujan’s Q-function. J. Comput. Appl. Math. 58, 103–116 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Koumandos, S.: A Bernstein function related to Ramanujan’s approximations of \(exp(n)\). Ramanujan J. 30, 447–459 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ramanujan, S.: Question 294. J. Indian Math. Soc. 3, 128 (1911)Google Scholar
  8. 8.
    Ramanujan, S.: Collected Papers. Chelsea, New York (1962)Google Scholar
  9. 9.
    Ruscheweyh, S., Salinas, L.: Universally prestarlike functions as convolution multipliers. Math. Z. 263, 607–617 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ruscheweyh, S., Salinas, L., Sugawa, T.: Completely monotone sequences and universally prestarlike functions. Israel J. Math. 171, 285–304 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Szegö, G.: Über einige von S. Ramanujan gestellte Aufgaben. J. Lond. Math. Soc. 3, 225–232 (1928)CrossRefzbMATHGoogle Scholar
  12. 12.
    Volkmer, H.: Factorial series connected with the Lambert function, and a problem posed by Ramanujan. Ramanujan J. 16, 235–245 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Watson, G.N.: Theorems stated by Ramanujan (V): approximations connected with \(e^{x}\). Proc. Lond. Math. Soc. 29, 293–308 (1929)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Andrew Bakan
    • 1
  • Stephan Ruscheweyh
    • 2
  • Luis Salinas
    • 3
  1. 1.Institute of MathematicsNational Academy of Sciences of UkraineKyivUkraine
  2. 2.Institut für MathematikUniversität WürzburgWürzburgGermany
  3. 3.Departamento de InformáticaUTFSMValparaísoChile

Personalised recommendations