Skip to main content
SpringerLink
Log in

Sharp power-type Heronian and Lehmer means inequalities for the complete elliptic integrals

  • Published:
Applied Mathematics-A Journal of Chinese Universities Aims and scope Submit manuscript

Abstract

In the article, we prove that the inequalities

$${H_p}({\cal K}(r),{\cal E}(r)) > {\pi \over 2},\,\,\,\,\,\,{L_q}({\cal K}(r),{\cal E}(r)) > {\pi \over 2}$$

hold for all r ∈ (0, 1) if and only if p ≥ −3/4 and q ≥ −3/4, where Hp(a, b) and Lq(a, b) are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b, and \({\cal K}(r)\) and \({\cal E}(r)\) are respectively the complete elliptic integrals of the first and second kinds.

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. Y M Chu, M K Wang, Y F Qiu. An optimal double inequality between power-type Heron and seiffert means, J Inequal Appl, 2010, 2010, Article ID: 146945.

  2. H Alzer. Über Lehmers Mittelwertfamilie, Elem Math, 1988, 43(2): 50–54.

    MathSciNet  MATH  Google Scholar 

  3. S S Zhou, W M Qian, Y M Chu, X H Zhang. Sharp power-type Heronian mean bounds for the Sándor and Yang means, J Inequal Appl, 2015, 2015: 159.

    Article  MATH  Google Scholar 

  4. Y M Chu, T H Zhao. Convexity and concavity of the complete elliptic integrals with respect to Lehmer mean, J Inequal Appl, 2015, 2015: 396.

    Article  MathSciNet  MATH  Google Scholar 

  5. Z Liu. Remark on inequalities between Hölder and Lehmer means, J Math Anal Appl, 2000, 247(1): 309–313.

    Article  MathSciNet  MATH  Google Scholar 

  6. M K Wang, Y F Qiu, Y M Chu. Sharp bounds for Seiffert means in terms of Lehmer means, J Math Inequal, 2010, 4(4): 581–586.

    Article  MathSciNet  MATH  Google Scholar 

  7. Y F Qiu, M K Wang, Y M Chu, G D Wang. Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean, J Math Inequal, 2011, 5(3): 301–306.

    Article  MathSciNet  MATH  Google Scholar 

  8. T H Zhao, Z Y He, Y M Chu. Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals, Comput Methods Funct Theory, 2020, 21: 413–426.

    Article  MATH  Google Scholar 

  9. M Abramowitz, I A Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, US Government Printing Office, Washington, 1964.

    MATH  Google Scholar 

  10. S Ponnusamy, M Vuorinen. Asymptotic expansions and inequalities for hypergeometric functions, Mathematika, 1997, 44(2): 278–301.

    Article  MathSciNet  MATH  Google Scholar 

  11. R Balasubramanian, S Ponnusamy, M Vuorinen. Functional inequalities for the quotients of hypergeometric functions, J Math Anal Appl, 1998, 218(1): 256–268.

    Article  MathSciNet  MATH  Google Scholar 

  12. R Balasubramanian, S Ponnusamy. On Ramanujan’s asymptotic expansions and inequalities for hypergeometric functions, Proc Indian Acad Sci (Math Sci), 1998, 10(2): 895–108.

    MathSciNet  MATH  Google Scholar 

  13. S Ponnusamy. Close-to-convexity properties of Gaussian hypergeometric functions, J Comput Appl Math, 1998, 88(2): 327–337.

    Article  MathSciNet  MATH  Google Scholar 

  14. W Becken, P Schmelcher. The analytic continuation of the Gaussian hypergeometric function2F1(a, b; c; z) for arbitrary parameters, J Comput Appl Math, 2000, 126(1–2): 449–478.

    Article  MathSciNet  MATH  Google Scholar 

  15. R Balasubramanian, S Naik, S Ponnusamy, M Vuorinen. Elliotts identity and hypergeometric functions, J Math Anal Appl, 2002, 271(1): 232–256.

    Article  MathSciNet  MATH  Google Scholar 

  16. A Baricz, S Ponnusamy, M Vuorinen. Functional inequalities for modified Bessel functions, Expo Math, 2011, 29(3): 399–414.

    Article  MathSciNet  MATH  Google Scholar 

  17. S Ponnusamy, M Vuorinen. Univalence and convexity properties for Gaussian hypergeometric functions, Rocky Mountain J Math, 2001, 31(1): 327–353.

    Article  MathSciNet  MATH  Google Scholar 

  18. J H Choi, Y C Kim, M Saigo. Geometric properties of convolution operators defined by Gaussian hypergeometric functions, Integral Transforms Spec Funct, 2002, 13(2): 117–130.

    Article  MathSciNet  MATH  Google Scholar 

  19. T H Zhao, Z Y He, Y M Chu. On some refinements for inequalities involving zero-balanced hypergeometric function, AIMS Mathematics, 2020, 5(6): 6479–6495.

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. S L Qiu, X Y Ma, Y M Chu. Sharp Landen transformation inequalities for hypergeometric functions, with applications, J Math Anal Appl, 2019, 474(2): 1306–1337.

    Article  MathSciNet  MATH  Google Scholar 

  25. M K Wang, Y M Chu, W Zhang. Monotonicity and inequalities involving zero-balanced hypergeometric function, Math Inequal Appl, 2019, 22(2): 601–617.

    MathSciNet  MATH  Google Scholar 

  26. T H Zhao, M K Wang, W Zhang, Y M Chu. Quadratic transformation inequalities for Gaussian hypergeometric function, J Inequal Appl, 2018, 2018: 251.

    Article  MathSciNet  MATH  Google Scholar 

  27. H Alzer. Sharp inequalities for the complete elliptic integral of the first kind, Math Proc Cambridge Philos Soc, 1998, 124(2): 309–314.

    Article  MathSciNet  MATH  Google Scholar 

  28. H Alzer, K Richards. A note on a function involving complete elliptic integrals: monotonicity, convexity, inequalities, Anal Math, 2015, 41(3): 133–139.

    Article  MathSciNet  MATH  Google Scholar 

  29. Z H Yang, J F Tian, Convexity and monotonicity for elliptic integrals of the first kind and applications, Appl Anal Discrete Math, 2019, 13(1): 240–260.

    Article  MathSciNet  MATH  Google Scholar 

  30. T H Zhao, M K Wang, Y M Chu. Monotonicity and convexity involving generalized elliptic integral of the first kind, Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM, 2021, 115(2).

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Z H Yang, Y M Chu, W Zhang. High accuracy asymptotic bounds for the complete elliptic integral of the second kind, Appl Math Comput, 2019, 348: 552–564.

    MathSciNet  MATH  Google Scholar 

  33. Z H Yang, W M Qian, Y M Chu, W Zhang. On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind, J Math Anal Appl, 2018, 462(2): 1714–1726.

    Article  MathSciNet  MATH  Google Scholar 

  34. Z H Yang. Sharp approximations for the complete elliptic integrals of the second kind by one-parameter means, J Math Anal Appl, 2018, 467(1): 446–461.

    Article  MathSciNet  MATH  Google Scholar 

  35. G D Anderson, M K Vamanamurthy, M Vuorinen. Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.

    MATH  Google Scholar 

  36. M K Wang, Y M Chu, Y F Qiu, S L Qiu. An optimal power mean inequality for the complete elliptic integrals, Appl Math Letters, 2011, 24(6): 887–890.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China

    Tie-hong Zhao

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

    Yu-ming Chu

Authors
  1. Tie-hong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu-ming Chu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu-ming Chu.

Ethics declarations

Conflict of interest The authors declare no conflict of interest.

Additional information

Supported by the National Natural Science Foundation of China(11971142) and the Natural Science Foundation of Zhejiang Province(LY19A010012)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, Th., Chu, Ym. Sharp power-type Heronian and Lehmer means inequalities for the complete elliptic integrals. Appl. Math. J. Chin. Univ. 38, 467–474 (2023). https://doi.org/10.1007/s11766-023-4223-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11766-023-4223-9

MR Subject Classification

Keywords

Search

Navigation