Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1221.26037
Liu, Hong; Meng, Xiang-Ju
The optimal convex combination bounds for Seiffert's mean.
(English)
[J] J. Inequal. Appl. 2011, Article ID 686834, 9 p. (2011). ISSN 1029-242X/e

The authors prove the following optimal bounds for the Seiffert mean $P(a,b)=(a-b)/[2\arcsin ((a-b)/(a+b))]$ by convex combinations of contraharmonic mean $C(a,b)=(a^{2}+b^{2})/(a+b)$ and geometric mean $G(a,b)= \sqrt{ab}$, respectively, harmonic mean $H(a,b)=2ab/(a+b)$. \par 1) The double inequality $\alpha _{1}C(a,b)+(1-\alpha _{1})G(a,b)<P(a,b)<\beta _{1}C(a,b)+(1-\beta _{1})G(a,b)$ holds for all $a,b>0$ with $a\neq b$ if and only if $\alpha _{1}\leq 2/9$ and $\beta _{1}\geq 1/\pi$. \par 2) The double inequality $\alpha _{2}C(a,b)+(1-\alpha _{2})H(a,b)<P(a,b)<\beta _{2}C(a,b)+(1-\beta _{2})H(a,b)$ holds for all $a,b>0$ with $a\neq b$ if and only if $\alpha _{2}\leq 1/\pi$ and $\beta _{2}\geq 5/12$.
MSC 2000:
*26E60 Means

Keywords: Seiffert's mean; contraharmonic mean; harmonic mean; geometric mean

Highlights
Master Server