Rad HAZU, Matematičke znanosti, Vol. 19 (2015), 129-142.

SEIFFERT MEANS, ASYMPTOTIC EXPANSIONS AND INEQUALITIES

Lenka Vukšić

Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia
e-mail: lenka.vuksic@fer.hr

Abstract.   In this paper we study inequalities of the form

(1 - μ) M1(s, t) + μ M3(s, t) ≤ M2(s, t) ≤ (1 - ν) M1(s, t) + ν M3(s, t),

which cover some classical bivariate means and Seiffert means. Using techniques of asymptotic expansions detailed analysis was made and the method for obtaining optimal parameters μ and ν was described.

2010 Mathematics Subject Classification.   26E60, 41A60.

Key words and phrases.   Seiffert means, asymptotic expansion, optimal convex combination.

Full text (PDF) (free access)

References:

  1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964.
    MathSciNet

  2. J. Aczél and Z. Páles, The behaviour of means under equal increments of their variables, Amer. Math. Monthly 95 (1988), 856-860.
    MathSciNet     CrossRef

  3. P. S. Bullen, Averages still on the move, Math. Mag. 63 (1990), 250-255.
    MathSciNet     CrossRef

  4. P. S. Bullen, Handbook of Means and Their Inequalities, Kluwer Academic Publishers, Dordrecht, 2003.
    MathSciNet     CrossRef

  5. P. S. Bullen, D. S. Mitrinović and P. M. Vasić, Means and their inequalities, D. Reidel Publishing Co., Dordrecht, 1988.
    MathSciNet     CrossRef

  6. C.-P. Chen, N. Elezović and L. Vukšić, Asymptotic formulae associated with the Wallis power function and digamma function, J. Class. Anal. 2 (2013), 151-166.
    MathSciNet

  7. Y. M. Chu, Y. F. Qiu, M. K. Wang and G. D. Wang, The Optimal convex combination bounds of arithmetic and harmonic means for the Seiffert's mean, J. Inequal. Appl. 2010, Art. ID 436457, 7 pp.
    MathSciNet     CrossRef

  8. Y. M. Chu, M. K. Wang and W. M. Gong, Two sharp double inequalities for Seiffert mean, J. Inequal. Appl. 2011, 2011:44, 7 pp.
    MathSciNet     CrossRef

  9. Y. M. Chu, C. Zong and G. D. Wang, Optimal convex combination bounds of Seiffert and geometric means for the arithmetic mean, J. Math. Inequal. 5 (2011), 429-434.
    MathSciNet     CrossRef

  10. N. Elezović, Asymptotic inequalities and comparison of classical means, J. Math. Inequal. 9 (2015), 177-196.
    MathSciNet     CrossRef

  11. N. Elezović and L. Vukšić, Asymptotic expansions of bivariate classical means and related inequalities, J. Math. Inequal. 8 (2014), 707-724.
    MathSciNet     CrossRef

  12. A. Erdélyi, Asymptotic expansions, Dover Publications, 1956.
    MathSciNet

  13. S.-Q. Gao, H.-Y. Gao and W.-Y. Shi, Optimal convex combination bounds of the centroidal and harmonic means for the Seiffert mean, Int. J. Pure Appl. Math. 70 (2011), 701-709.

  14. L. Hoehn and I. Niven, Averages on the move, Math. Mag. 58 (1985), 151-156.
    MathSciNet     CrossRef

  15. W.-D. Jiang and J. Cao and F. Qi, Two sharp inequalities for bounding the Seiffert mean by the arithmetic, centroidal, and contra-harmonic means,
    arXiv:1201.6432v1 [math.CA]

  16. W.-D. Jiang and F. Qi, Some sharp inequalities involving Seiffert and other means and their concise proofs, Math. Inequal. Appl. 15 (2012), 1007-1017.
    MathSciNet     CrossRef

  17. H. Liu and X. J. Meng, The optimal convex combination bounds for Seiffert's mean, J. Inequal. Appl. 2011, Art. ID 686834, 9 pp.
    MathSciNet     CrossRef

  18. E. Neuman, A one-parameter family of bivariate means, J. Math. Inequal. 7 (2013), 399-412.
    MathSciNet     CrossRef

Rad HAZU Home Page