Logarithmic mean and weighted sum of geometric and anti-harmonic means

Authors

  • Mira Cristiana Anisiu Tiberiu Popoviciu Institute of Numerical Analysis, Romania
  • Valeriu Anisiu Babeş-Bolyai University, Cluj-Napoca, Romania

DOI:

https://doi.org/10.33993/jnaat412-971

Keywords:

two-variable means, weighted arithmetic mean, inequalities, symbolic computer algebra
Abstract views: 308

Abstract

We consider the problem of finding the optimal values \(\alpha,\ \beta\in\mathbb{R}\) for which the inequality\[\alpha G(a,b)+(1-\alpha)C(a,b)<L(a,b)<\beta G(a,b)+(1-\beta)C(a,b)\]holds for all \(a,b>0\), \(a\neq b\), where \(G(a,b),L(a,b)\) and \(C(a,b)\) are respectively the geometric, logarithmic and anti-harmonic means of \(a\) and \(b\).

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References

H. Alzer and S.-L. Qiu, Inequalities for means in two variables, Arch. Math. (Basel), 80 (2003), pp. 201-215, https://doi.org/10.1007/s00013-003-0456-2 DOI: https://doi.org/10.1007/s00013-003-0456-2

M.-C. Anisiu and V. Anisiu, Bilateral inequalities for means, communication, 9th Joint Conference on Mathematics and Computer Science (MACS 2012), Siófok, Hungary, February 9-12, Abstracts, p. 15, 2012.

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, Article ID 436457, 2010, 7 pages, https://doi.org/10.1155/2010/436457 DOI: https://doi.org/10.1155/2010/436457

W.-F. Xia and Y.-M. Chu, Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means, Rev. Anal. Numér. Théor. Approx., 39 (2010) no. 2, pp. 176-183, http://ictp.acad.ro/jnaat/journal/article/view/2010-vol39-no2-art10

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Published

2012-08-01

How to Cite

Anisiu, M. C., & Anisiu, V. (2012). Logarithmic mean and weighted sum of geometric and anti-harmonic means. Rev. Anal. Numér. Théor. Approx., 41(2), 95–98. https://doi.org/10.33993/jnaat412-971

Issue

Vol. 41 No. 2 (2012)

Section

Articles

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Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory

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