*(English)*Zbl 1053.26015

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

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

if $0\le x<y$ and

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., $SB(\frac{x+y}{2},\sqrt{xy})$ 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.