Bilateral inequalities for 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/jnaat422-985

Keywords:

two-variable means, weighted arithmetic mean, inequalities, symbolic computer algebra
Abstract views: 215

Abstract

Let \(\left(M_{1},M_{2},M_{3}\right) \) be three means in two variables chosen from \(H\), \(G\), \(L\), \(I\), \(A\), \(Q\), \(S\), \(C\) so that \[ M_{1}(a,b)<M_{2}(a,b)<M_{3}(a,b),\quad 0<a<b. \] We consider the problem of finding \(\alpha,\ \beta\in\mathbb{R}\) for which \[ \alpha M_{1}(a,b)+(1-\alpha)M_{3}(a,b)<M_{2}(a,b)<\beta M_{1}(a,b)+(1-\beta )M_{3}(a,b). \] We solve the problem for the triplets \(\left(G,L,A\right)\), \(\left(G,A,Q\right)\), \(\left(G,A,C\right)\), \(\left(G,Q,C\right)\), \(\left(A,Q,C\right) \), \(\left(A,S,C\right) \), \(\left(A,Q,S\right) \) and \((L,A,C)\). The Symbolic Algebra Program Maple is used to determine the range where some parameters can vary, or to find the minimal polynomial for an algebraic number.

Downloads

Download data is not yet available.

References

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

H. Alzer and S. Ruscheweyh, On the intersection of two-parameter mean value families, Proc. A. M. S., 129(9) (2001), pp. 2655-2662. DOI: https://doi.org/10.1090/S0002-9939-01-06050-6

M. C. Anisiu and V. Anisiu, Refinement of some inequalities for means, Rev. Anal. Numér. Théor. Approx., 35 (2006) no. 1, pp.5—10, http://ictp.acad.ro/jnaat/journal/article/view/2006-vol35-no1-art2

M. C. Anisiu and V. Anisiu, Logarithmic mean and weighted sum of geometric and anti-harmonic means, Rev. Anal. Numér. Théor. Approx., 41 (2012) no. 2, pp. 95-98, http://ictp.acad.ro/jnaat/journal/article/view/2012-vol41-no2-art1

P. S. Bullen, Handbook of Means and Their Inequalities, Series: Mathematics and Its Applications, vol. 560, 2nd ed., Kluwer Academic Publishers Group, Dordrecht, 2003.

C. Gini, Di una formula comprensiva delle medie, Metron, 13 (1938), pp. 3-22.

M. Ivan and I. Raşa, Some inequalities for means, Tiberiu Popoviciu Itinerant Seminar of Functional Equations, Approximation and Convexity, Cluj-Napoca, May 23-29, 2000, pp. 99-102.

K. B. Stolarsky, Generalizations of the logarithmic mean, Math. Mag., 48 (1975), pp. 87--92. DOI: https://doi.org/10.1080/0025570X.1975.11976447

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

Downloads

Published

2013-08-01

How to Cite

Anisiu, M.-C., & Anisiu, V. (2013). Bilateral inequalities for means. Rev. Anal. Numér. Théor. Approx., 42(2), 94–102. https://doi.org/10.33993/jnaat422-985

Issue

Vol. 42 No. 2 (2013)

Section

Articles

License

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

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.