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Monotonicity properties for generalized logarithmic means. (English) Zbl 1063.26025

The generalized logarithmic mean \(L_{r}(a,b)\) of two positive numbers \(a\neq b\) is defined by \(L_{r}(a,b)=\left( \frac{b^{r+1}-a^{r+1}}{\left( r+1\right) \left( b-a\right) }\right) ^{1/r},\) for \(r\neq -1,0\), while \(L_{-1},L_{0}\) are, respectively, the logarithmic and the identric mean. In this paper, the monotonicity of the ratio \(L_{r}(a,x)/L_{s}(a,x)\) is studied.

MSC:

26E60 Means
26A48 Monotonic functions, generalizations
26D07 Inequalities involving other types of functions
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