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)
On the Schwab-Borchardt mean. (English) Zbl 1053.26015

Given two positive numbers x,y, the Gaussian iteration

x 0 =x,y 0 =y,x n+1 =x n +y n 2,y n+1 =x n+1 y n

converges to the Schwab-Borchardt mean SB(x,y) of x,y which can be expressed explicitely as

SB(x,y)=y 2 -x 2 arccos(x/y)

if 0x<y and

SB(x,y)=x 2 -y 2 arcosh(x/y)

if 0y<x. This mean is homogeneous but nonsymmetric. Due to various representations of this mean, if x and y are replaced by the arithmetic, geometric and quadratic means of x and y one obtains various classical two variable means, e.g., SB(x+y 2,xy) results the logarithmic mean. The so-called Seiffert means can also be obtained this way.

The main results of the paper offer comparison and Ky Fan type inequalities for the Schwab-Borchardt mean, logarithmic mean, the Seiffert-type means, and the Gauss arithmetic-geometric mean. The sequential method of Sándor is generalized to obtain bounds for the means under discussion.

26D15Inequalities for sums, series and integrals of real functions