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)
An optimal double inequality between power-type Heron and Seiffert means. (English) Zbl 1210.26021

Purpose of this paper is to present the optimal upper and lower power-type Heron mean bounds for the Seiffert mean T(a,b).

For k[0;+), the power-type Heron mean H k (a,b) and the Seiffert mean T(a,b) of two positive real numbers a and b are defined by:

H k (a,b)=( a k + (ab) k/2 + b k ) / 3 1/k ,k0,ab,k=0,


T(a,b)=(a-b)/2arctan( a - b ) / ( a + b ),ab,a,a=b,


It is proved that for all a,b>0, with ab, one has:

H log3/log(π/2) (a,b)<T(a,b)<H 5/2 (a,b)

and both H’s are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean T(a,b), respectively.

26D15Inequalities for sums, series and integrals of real functions