Skip to main content
SpringerLink
Log in

A Note on the Hypergeometric Mean Value

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

Recent efforts to obtain bounds for the complete elliptic integral

$${\pi \over 2} \cdot {_2F_1} \bigg(-{1 \over 2},{1 \over 2}; 1;r^{2}\bigg)$$

in terms of power means and other related means have precipitated the search for similar bounds for the more general 2 F 1(α,β;γ;r). In an early paper, B. C. Carlson considered the approximation of the hypergeometric mean values (2 F 1(−a,b;b + c;r))1/a in terms of means of order t, given by M t(s,r):= (1 − s) + s(1 − r)t 1/t. In this note, a refinement of one such approximation is established by first proving a general positivity result involving 3 F 2.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Almkvist and B. Berndt, Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary, Amer. Math. Monthly, 95 (1988), 585–608.

    Article  MathSciNet  MATH  Google Scholar 

  2. Horst Alzer, personal correspondence, 1998.

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

    MATH  Google Scholar 

  4. G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, 1999.

  5. R. Askey, Orthogonal Polynomials and Special Functions, SIAM, Philadelphia, 1975.

    Book  Google Scholar 

  6. R. Askey, G. Gasper, and M. Ismail, A positive sum from summability theory, J. Approx. Theory 13 (1975), 413–420.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. W. Barnard, K. Pearce, and K. C. Richards, An inequality involving the generalized hypergeometric function and the arc length of an ellipse, SIAM J. Math. Anal. 31 (2000), 693–699.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. W. Barnard, K. Pearce, and K. C. Richards, A monotonicity property involving 3 F 2 and comparisons of classical approximations of elliptical arc length, SIAM J. Math. Anal. 32 (2000), 403–419.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. F. Beckenbach and R. Bellman, Inequalities, fourth printing, Springer-Verlag, Berlin, 1983.

    Google Scholar 

  10. J. Borwein and P. Borwein, Pi and the AGM, Wiley, New York, 1987.

    MATH  Google Scholar 

  11. J. Borwein and P. Borwein, Inequalities for compound mean iterations with logarithmic asymptotes, J. Math. Anal. Appl. 177 (1993), 572–582.

    Article  MathSciNet  MATH  Google Scholar 

  12. B. C. Carlson, Some inequalities for hypergeometric functions, Proc. Amer. Math. Soc. 16 (1966), 32–39.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, 79409-1042, USA

    Roger W. Barnard

  2. Department of Mathematics, Southwestern University, Georgetown, Texas, 78626, USA

    Kendall C. Richards

Authors
  1. Roger W. Barnard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kendall C. Richards
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roger W. Barnard.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barnard, R.W., Richards, K.C. A Note on the Hypergeometric Mean Value. Comput. Methods Funct. Theory 1, 81–88 (2001). https://doi.org/10.1007/BF03320978

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03320978

Keywords

2000 MSC

Search

Navigation