Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means
DOI:
https://doi.org/10.33993/jnaat392-1038Keywords:
logarithmic mean, identric mean, arithmetic mean, harmonic meanAbstract
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
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 DOI: 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 DOI: 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 DOI: 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. DOI: https://doi.org/10.1007/978-94-017-2226-1
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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: https://doi.org/10.1080/0025570X.1975.11976447
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 DOI: 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 DOI: 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 DOI: 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 DOI: https://doi.org/10.1080/00029890.1985.11971549
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951. DOI: https://doi.org/10.1515/9781400882663
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 DOI: 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 DOI: https://doi.org/10.1007/s00013-003-0456-2
Published
How to Cite
Issue
Section
License
Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory
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.
