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 1187.26016
Long, Bo-Yong; Chu, Yu-Ming
Optimal power mean bounds for the weighted geometric mean of classical means.
(English)
[J] J. Inequal. Appl. 2010, Article ID 905679, 6 p. (2010). ISSN 1029-242X/e

Summary: For $p\in\Bbb R$, the power mean of order $p$ of two positive numbers $a$ and $b$ is defined by $M_p(a,b)= ((a^p+b^p)/2)^{1/p}$, for $p\ne 0$, and $M_p(a,b)= \sqrt{ab}$, for $p=0$. In this paper, we answer the question: what are the greatest value $p$ and the least value $q$ such that the double inequality $M_p(a,b)\le A^\alpha(a,b) G^\beta(a,b) H^{1-\alpha-\beta}(a,b)\le M_q(a,b)$ holds for all $a,b>0$ and $\alpha,\beta>0$ with $\alpha+\beta<1$? Here $A(a,b)= (a+b)/2$, $G(a,b)= \sqrt{ab}$, and $H(a,b)= 2ab/(a+b)$ denote the classical arithmetic, geometric, and harmonic means, respectively.
MSC 2000:
*26E60 Means
26D15 Inequalities for sums, series and integrals of real functions

Highlights
Master Server