zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
A note on some inequalities for means. (English) Zbl 0693.26005
The logarithmic and identric means of two positive numbers a and b are defined by $L=L(a,b):=(b-a)/(\ln b-\ln a)$ for $a\ne b;\quad L(a,a)=a,$ and $I=I(a,b):=\frac{1}{e}(b\sp b/a\sp a)\sp{1/(b-a)}$ for $a\ne b,\quad I(a,a)=a,$ respectively. Let $A=A(a,b):=(a+b)/2$ and $G=G(a,b):=\sqrt{ab}$ denote the arithmetic and geometric means of a and b, respectively. Recently, in two interesting papers, H. Alzer has obtained the following inequalities: $(1)\quad A.G<L.I$ and $L+I<A+G;\quad (2)\quad \sqrt{G.I}<L<\frac{1}{2}(G+I)$ which hold true for all positive $a\ne b.$ In our paper we prove, by using new methods, that the left side of (1) is weaker than the left side of (2), while the right side of (1) is stronger than the right side of (2).
Reviewer: J.Sándor

26D15Inequalities for sums, series and integrals of real functions
Full Text: DOI
[1] H. Alzer, Ungleichungen f?r Mittelwerte. Arch. Math.47, 422-426 (1986). · Zbl 0585.26014 · doi:10.1007/BF01189983
[2] H. Alzer, Two inequalities for means. C.R. Math. Rep. Acad. Sci. Canada9, 11-16 (1987). · Zbl 0615.26015
[3] B. C. Carlson, The logarithmic mean. Amer. Math. Monthly79, 615-618 (1972). · Zbl 0241.33001 · doi:10.2307/2317088
[4] K. S.Kunz, Numerical analysis. New York 1957. · Zbl 0079.33601
[5] E. B. Leach andM. C. Sholander, Extended mean values II. J. Math. Anal. Appl.92, 207-223 (1983). · Zbl 0517.26007 · doi:10.1016/0022-247X(83)90280-9
[6] T. P. Lin, The power mean and the logarithmic mean. Amer. Math. Monthly81, 879-883 (1974). · Zbl 0292.26015 · doi:10.2307/2319447
[7] J. S?ndor, Some integral inequalities. Elem. Math.43, 177-180 (1988).
[8] J.S?ndor, Inequalities for means. Submitted.
[9] H. J. Seiffert, Eine Integralungleichung f?r streng monotone Funktionen mit logarithmisch konvexer Umkehrfunktion. Elem. Math.44, 16-18 (1989). · Zbl 0721.26010
[10] K. B. Stolarsky, The power and generalized means. Amer. Math. Monthly.87, 545-548 (1980). · Zbl 0455.26008 · doi:10.2307/2321420