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

Given two positive numbers $x,y$, the Gaussian iteration

${x}_{0}=x,\phantom{\rule{1.em}{0ex}}{y}_{0}=y,\phantom{\rule{1.em}{0ex}}{x}_{n+1}=\frac{{x}_{n}+{y}_{n}}{2},\phantom{\rule{1.em}{0ex}}{y}_{n+1}=\sqrt{{x}_{n+1}{y}_{n}}$

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

$SB\left(x,y\right)=\frac{\sqrt{{y}^{2}-{x}^{2}}}{arccos\left(x/y\right)}$

if $0\le x and

$SB\left(x,y\right)=\frac{\sqrt{{x}^{2}-{y}^{2}}}{\text{arcosh}\left(x/y\right)}$

if $0\le y. 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., $SB\left(\frac{x+y}{2},\sqrt{xy}\right)$ 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.

##### MSC:
 26D15 Inequalities for sums, series and integrals of real functions 26E60 Means