zbMATH — the first resource for mathematics

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

26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
26A48 Monotonic functions, generalizations
Full Text: DOI EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.