The monotonicity of the ratio between generalized logarithmic means (Q930978)

The generalized logarithmic mean \(L_{r}(a,b)\) of two positive numbers \(a,b\) is defined by \(L_{r}(a,a)=a\) and if \(a\neq b\), by \[ L_{r}(a,b)=\left( \frac{b^{r+1}-a^{r+1}}{(r+1)(b-a)}\right) ,\;r\neq0,1;\;L_{-1}(a,b)=\frac{b-a}{\ln b-\ln a},\;L_{0}(a,b)=\frac{1}{e}\left( \frac{b^{b}}{a^{a}}\right) ^{\frac{1}{b-a}}. \] In this paper the following properties are proven: Let \(\delta,\varepsilon\) and \(b\) be positive numbers, then the function \[ \frac{L_{r}(b,b+\varepsilon)}{L_{r}(b+\delta,b+\delta+\varepsilon)} \] is strictly increasing in \(r\in(-\infty,\infty).\) Let \(n\) be a natural number, then, for all \(r\in\mathbb{R}\), \[ \frac{n}{n+1}<\left( \frac{\frac{1}{n}\sum_{i=1}^{n}i^{r}}{\frac{1}{n+1} \sum_{i=1}^{n+1}i^{r}}\right) ^{1/r}<1. \] Moreover, the lower bound \(\frac{n}{n+1}\) and the upper bound \(1\) are the best possible.