zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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) (ab),I(a,a)=a, the logarithmic mean L(a,b)=(b-a)ln -1 (b/a)(ab),L(a,a)=a(a>0,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(ab,t0)·

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)(=e(b a a -b ) 1/(a-b) forba,J(a,a)=a)·

Reviewer: J.Aczél

MSC:
26D15Inequalities for sums, series and integrals of real functions
26A51Convexity, generalizations (one real variable)
26A48Monotonic functions, generalizations (one real variable)
References:
[1]Alzer, H.,Two inequalities for means. C.R. Math. Rep. Acad. Sci. Canada.9 (1987), 11–16.
[2]Alzer, H.,Ungleichungen für Mittelwerte. Arch. Math. (Basel)47 (1986), 422–426.
[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. · 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.
[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.
[13]Sándor, J.,Some integral inequalities. Elem. Math.43 (1988), 177–180.
[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.
[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