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On the Schwab-Borchardt mean. (English) Zbl 1053.26015
Given two positive numbers $$x,y$$, the Gaussian iteration $x_0=x,\quad y_0=y,\quad x_{n+1}={x_n+y_n\over2}, \quad y_{n+1}=\sqrt{x_{n+1}y_n}$ 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 $$0\leq x<y$$ and $SB(x,y)={\sqrt{x^2-y^2}\over\text{arcosh}{(x/y)}}$ if $$0\leq y<x$$. 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.
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 26E60 Means
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