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.


26D15 Inequalities for sums, series and integrals
26E60 Means
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