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The Schur geometrical convexity of the extended mean values. (English) Zbl 1163.26004
The extended mean value \(E(r,s;x,y)\) is a function of \(r,s\in \mathbb{R}\) and \(x,y>0\) which includes, for various values of \(r\) and \(s\), many well-known mean values of two positive numbers \(x\), \(y\): arithmetic mean, geometric mean, harmonic mean, Hölder mean, logarithmic mean etc. The main result of the paper is that \(E(r,s;\cdot,\cdot)\) is Schur geometrically convex (resp., concave) on \((0,+\infty)\times(0,\infty)\) if and only if \(s+r\geq0\) (resp., \(r+s\leq0\)).

MSC:
26B25 Convexity of real functions of several variables, generalizations
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