A separation of some Seiffert-type means by power means

Iulia Costin; Gheorghe Toader

doi:10.33993/jnaat412-974

Authors

Iulia Costin Technical University of Cluj-Napoca, Romania
Gheorghe Toader Technical University of Cluj-Napoca, Romania

DOI:

https://doi.org/10.33993/jnaat412-974

Keywords:

Seiffert type means, power means, logarithmic mean, identric mean, inequalities of means

Abstract views: 231

Abstract

Consider the identric mean \(\mathcal{I}\), the logarithmic mean \(\mathcal{L,}\) two trigonometric means defined by H. J. Seiffert and denoted by \(\mathcal{P}\) and \(\mathcal{T,}\) and the hyperbolic mean \(\mathcal{M}\) defined by E. Neuman and J. Sándor. There are a number of known inequalities between these means and some power means \(\mathcal{A}_{p}.\) We add to these inequalities some new results obtaining the following chain of inequalities\[\mathcal{A}_{0}<\mathcal{L}<\mathcal{A}_{1/3}<\mathcal{P<A}_{2/3}<\mathcal{I}<\mathcal{A}_{3/3}<\mathcal{M}<\mathcal{A}_{4/3}<\mathcal{T}<\mathcal{A}_{5/3}.\]

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References

B.C. Carlson, The logarithmic mean, Amer. Math. Monthly, 79 (1972), pp. 615-618, https://doi.org/10.1080/00029890.1972.11993095 DOI: https://doi.org/10.1080/00029890.1972.11993095

I. Costin and G. Toader, A nice evaluation of some Seiffert type means by power means, Int. J. Math. Math. Sc., 2012, Article ID 430692, 6 pages, https://doi.org/10.1155/2012/430692 DOI: https://doi.org/10.1155/2012/430692

P.A. Hästö, Optimal inequalities between Seiffert's means and power means, Math. Inequal. Appl., 7 (2004) no. 1, pp. 47-53, https://doi.org/10.7153/mia-07-06 DOI: https://doi.org/10.7153/mia-07-06

A.A. Jagers, Solution of problem 887, Niew Arch. Wisk. (Ser. 4), 12 (1994), pp. 230-231, https://doi.org/10.1016/0734-9750(94)90846-x DOI: https://doi.org/10.1016/0734-9750(94)90846-X

T.P. Lin, The power and the logarithmic mean, Amer. Math. Monthly, 81 (1974), pp. 879-883, https://doi.org/10.1080/00029890.1974.11993684 DOI: https://doi.org/10.1080/00029890.1974.11993684

E. Neuman and J. Sándor, On the Schwab-Borchardt mean, Math. Panon., 14 (2003) no. 2, pp. 253-266.

E. Neuman and J. Sándor, Companion inequalities for certain bivariate means, Appl. Anal. Discrete Math., 3 (2009), pp. 46-51, https://doi.org/10.2298/aadm0901046n DOI: https://doi.org/10.2298/AADM0901046N

A.O. Pittenger, Inequalities between arithmetic and logarithmic means, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 678-715 (1980), pp. 15-18.

H.-J. Seiffert, Problem 887, Niew Arch. Wisk. (Ser. 4), 11 (1993), pp. 176-176. DOI: https://doi.org/10.1080/02640419308729981

H.-J. Seiffert, Aufgabe β16, Die Wurzel, 29 (1995), pp. 221-222.