Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means

Authors

  • Wei-feng Xia Huzhou Teachers College
  • Chu Yu-Ming Huzhou Teachers College

Keywords:

logarithmic mean, identric mean, arithmetic mean, harmonic mean

Abstract

The logarithmic mean \(L(a,b)\), identric mean \(I(a,b)\), arithmeticmean \(A(a,b)\) and harmonic mean \(H(a,b)\) of two positive real values \(a\) and \(b\) are defined by\begin{align*}\label{main}&L(a,b)=\begin{cases}\tfrac{b-a}{\log b-\log a},& a\neq b,\\a,&a=b,\end{cases}\\&I(a,b)=\begin{cases}\tfrac{1}{{\rm e}}\left(\tfrac{b^b}{a^a}\right)^{\tfrac{1}{b-a}},& a\neq b,\\a,&a=b,\end{cases}\end{align*}\(A(a,b)=\tfrac{a+b}{2}\) and \(H(a,b)=\tfrac{2ab}{a+b}\), respectively. In this article, we answer the questions: What are the best possible parameters \(\alpha_{1},\alpha_{2},\beta_{1}\) and \(\beta_{2}\), such that \(\alpha_{1}A(a,b)+(1-\alpha_{1})H(a,b)\leq L(a,b)\leq\beta_{1}A(a,b)+(1-\beta_{1})H(a,b)\) and \(\alpha_{2}A(a,b)+(1-\alpha_{2})H(a,b)\leq I(a,b)\leq\beta_2A(a,b)+(1-\beta_{2})H(a,b)\) hold for all \(a,b>0\)?

Downloads

Download data is not yet available.

References

M. Tominaga, Specht's ratio and logarithmic mean in the Young inequality, Math. Inequal. Appl., 7(1), pp. 113-125, 2004, https://doi.org/10.7153/mia-07-13

F. Qi and B.N. Guo, An inequality between ratio of the extended logarithmic means and ratio of the exponential means, Taiwanese J. Math., 7(2), pp. 229-237, https://doi.org/10.11650/twjm/1500575060

J. Maloney, J. Heidel and J. Pečarić, A reverse Hölder type inequality for the logarithmic mean and generalizations, J. Austral. Math. Soc. Ser. B, 41(3), pp. 401-409, 2000, https://doi.org/10.1017/s0334270000011322

A.O. Pittenger, The symmetric, logarithmic and power means, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 678-715, pp. 19-23, 1980.

P.S. Bullen, D.S. Mitrinović and P.M. Vasić, Means and Their inequalities, Dordrecht: D. Reidel Publishing Co., 1988.

H. Alzer, Ungleichungen für (e/a)a(b/e)b , Elem. Math., 40, pp. 120-123, 1985.

H. Alzer, Ungleichungen für Mittelwerte, Arch. Math. (Basel), 47(5), pp. 422-426, 1986, https://doi.org/10.1007/bf01189983

F. Burk, Notes: The geometric, logarithmic, and arithmetic mean inequality, Amer. Math. Monthly, 94(6), pp. 527-528, 1987, https://doi.org/10.1080/00029890.1987.12000678

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

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

J. Sándor, On the identric and logarithmic means, Aequationes Math., 40(2-3), pp. 261-270, 1990, https://doi.org/10.1007/bf02112299

J. Sándor, A note on some inequalities for means, Arch. Math. (Basel), 56(5), pp. 471-473, 1991, https://doi.org/10.1007/bf01200091

J. Sándor, On certain inequalities for means, J. Math. Anal. Appl., 189(2), pp. 602-606, 1995, https://doi.org/10.1006/jmaa.1995.1038

J. Sándor, On refinements of certain inequalities for means, Arch. Math. (Brno), 31(4), pp. 279-282, 1995.

J. Sándor, On certain inequalities for means II, J. Math. Anal. Appl., 199(2), pp. 629-635, 1996, https://doi.org/10.1006/jmaa.1996.0165

J. Sándor, On certain inequalities for means III, Arch. Math. (Basel), 76(1), pp. 34-40, 2001, https://doi.org/10.1007/s000130050539

J. Sándor and I. Rasa, Inequalities for certain means in two arguments, Nieuw Arch. Wisk. (4), 15(1-2), pp. 51-55, 1997, https://doi.org/10.1186/s13660-015-0828-8

J. Sándor and T. Trif, Some new inequalities for means of two arguments, Int. J. Math. Math. Sci., 25(8), pp. 525-532, 2001, https://doi.org/10.1155/s0161171201003064

O. Kouba, New bounds for the identric mean of two arguments, J. Inequal. Pure Appl. Math., 9(3), Article 71, 6 pp, 2008.

J. Chen and B. Hu, The identric mean and the power mean inequalities of Ky Fan type, Facta Univ. Ser. Math. Inform., 4, pp. 15-18, 1989.

H.J. Seiffert, Ungleichungen für einen bestimmten Mittelwert, Nieuw Arch. Wisk. (4), 13(2), pp. 195-198, 1995.

H.J. Seiffert, Ungleichungen für elementare Mittelwerte, Arch. Math. (Basel), 64(2), pp. 129-131, 1995, https://doi.org/10.1007/bf01196631

K.B. Stolarsky, Generalizations of the logarithmic mean, Math. Mag., 48, pp. 87-92, 1975,https://doi.org/10.2307/2689825

K.B. Stolarsky, The power and generalized logarithmic means, Amer. Math. Monthly, 87(7), pp. 545-548, 1980, https://doi.org/10.1080/00029890.1980.11995086

M.K. Vamanamurthy and M. Vuorinen, Inequalities for means, J. Math. Anal. Appl., 183(1), pp. 155-166, 1994, https://doi.org/10.1006/jmaa.1994.1137

P. Kahlig and J. Matkowski, Functional equations involving the logarithmic mean, Z. Angew. Math. Mech., 76(7), pp. 385-390, 1996, https://doi.org/10.1002/zamm.19960760710

A.O. Pittenger, The logarithmic mean in n variables, Amer. Math. Monthly, 92(2), pp. 99-104, 1985, https://doi.org/10.1080/00029890.1985.11971549

G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951.

E.B. Leach and M.C. Sholander, Extended mean values II, J. Math. Anal. Appl., 92(1), pp. 207-223, 1983, https://doi.org/10.1016/0022-247x(83)90280-9

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

Downloads

Published

2010-08-01

How to Cite

Xia, W.- feng, & Yu-Ming, C. (2010). Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means. Rev. Anal. Numér. Théor. Approx., 39(2), 176–183. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2010-vol39-no2-art10

Issue

Section

Articles