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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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