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Zbl 1053.26015
Neuman, E.; Sándor, J.
On the Schwab-Borchardt mean.
(English)
[J] Math. Pannonica 14, No. 2, 253-266 (2003). ISSN 0865-2090

Given two positive numbers $x,y$, the Gaussian iteration $$x_0=x,\quad y_0=y,\quad x_{n+1}={x_n+y_n\over2}, \quad y_{n+1}=\sqrt{x_{n+1}y_n}$$ converges to the Schwab-Borchardt mean $SB(x,y)$ of $x,y$ which can be expressed explicitely as $$SB(x,y)={\sqrt{y^2-x^2}\over\arccos{(x/y)}}$$ if $0\le x<y$ and $$SB(x,y)={\sqrt{x^2-y^2}\over\text{arcosh}{(x/y)}}$$ if $0\le y<x$. 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., $SB({x+y\over2},\sqrt{xy})$ results the logarithmic mean. The so-called Seiffert means can also be obtained this way. \par 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.
[Zsolt Páles (Debrecen)]
MSC 2000:
*26D15 Inequalities for sums, series and integrals of real functions
26E60 Means

Keywords: Schwab-Borchardt mean; Seiffert means; inequalities; Ky Fan type inequalities; logarithmic mean; arithmetic mean; geometric mean

Cited in: Zbl 1100.26011

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