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On the identric and logarithmic means. (English) Zbl 0717.26014
After a survey of existing results, several new ones are offered for the identric mean \(I(a,b)=e^{-1}(a^{-a}b^ b)^{1/(b-a)}\quad (a\neq b),\quad I(a,a)=a,\) the logarithmic mean \(L(a,b)=(b-a)\ln^{- 1}(b/a)\quad (a\neq b),\quad L(a,a)=a\quad (a>0,\quad b>0)\) and the arithmetic and geometric mean; for instance \[ L(a,b)I(a,b)^{t- 1}<L(a,b)(b^ t-a^ t)/(t(b-a))<(a^ t+b^ t)/2\quad (a\neq b,\quad t\neq 0). \] Logarithmic convexity and integral representations of the above means are used.
The definition of a “new mean” is unfortunately misprinted: it should be \[ J(a,b):=1/I(1/a,1/b)\quad (=\quad e(b^ aa^{-b})^{1/(a- b)}\text{ for } b\neq a,\quad J(a,a)=a). \]
Reviewer: J.Aczél

MSC:
26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
26A48 Monotonic functions, generalizations
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References:
[1] Alzer, H.,Two inequalities for means. C.R. Math. Rep. Acad. Sci. Canada.9 (1987), 11–16. · Zbl 0615.26015
[2] Alzer, H.,Ungleichungen für Mittelwerte. Arch. Math. (Basel)47 (1986), 422–426. · Zbl 0585.26014
[3] Alzer, H.,On an inequality of Ky Fan. J. Math. Anal. Appl.137 (1989), 168–172. · Zbl 0668.26012 · doi:10.1016/0022-247X(89)90280-1
[4] Beckenbach, E. F. andBellman, R.,Inequalities. Springer, New York, 1965.
[5] Carlson, B. C.,Some inequalities for hypergeometric functions. Proc. Amer. Math. Soc.17 (1966), 32–39. · Zbl 0137.26803 · doi:10.1090/S0002-9939-1966-0188497-6
[6] Carlson, B. C.,The logarithmic mean. Amer. Math. Monthly79 (1972), 615–618. · Zbl 0241.33001 · doi:10.2307/2317088
[7] Hardy, G. H., Littlewood, J. E. andPolya, G.,Inequalities. Cambridge Univ. Press, Cambridge–New York, 1988.
[8] Leach, E. B. andSholander, M. C.,Extended mean values II. J. Math. Anal. Appl.92 (1983), 207–223. · Zbl 0517.26007 · doi:10.1016/0022-247X(83)90280-9
[9] Lin, T. P.,The power mean and the logarithmic mean. Amer. Math. Monthly81 (1974), 879–883. · Zbl 0292.26015 · doi:10.2307/2319447
[10] Mitrinovic, D. S. (in cooperation withP. M. Vasic),Analytic Inequalities. Springer, Berlin–Heidelberg–New York, 1970. · Zbl 0199.38101
[11] Ostle, B. andTerwilliger, H. L.,A comparison of two means. Proc. Montana Acad. Sci.17 (1957), 69–70.
[12] Rüthing, D.,Eine allgemeine logarithmische Ungleichung. Elem. Math.41 (1986), 14–16. · Zbl 0607.26010
[13] Sándor, J.,Some integral inequalities. Elem. Math.43 (1988), 177–180. · Zbl 0702.26016
[14] Sándor, J.,An application of the Jensen – Hadamard inequality. To appear in Nieuw Arch. Wisk. (4)8 (1990).
[15] Sándor, J.,On an inequality of Ky Fan. To appear in Sem. Math. Anal., Babes–Bolyai Univ.
[16] Seiffert, H.-J.,Eine Integralungleichung für streng monotone Funktionen mit logarithmische konvexer Umkehrfunktion. Elem. Math.44 (1989), 16–17. · Zbl 0721.26010
[17] Stolarsky, K. B.,Generalizations of the logarithmic mean. Math. Mag.48 (1975), 87–92. · Zbl 0302.26003 · doi:10.2307/2689825
[18] Stolarsky, K. B.,The power and generalized logarithmic means. Amer. Math. Monthly87 (1980), 545–548. · Zbl 0455.26008 · doi:10.2307/2321420
[19] Zaiming, Z.,Problem E 3142. Amer. Math. Monthly93 (1986), 299. · doi:10.2307/2323689
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