The monotonicity of the ratio between generalized logarithmic means. (English) Zbl 1160.26012

Chen, Chao-Ping

J. Math. Anal. Appl. 345, No. 1, 86-89 (2008).

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.

Reviewer: Joaquím Martín (Barcelona)

MSC:

26D15

Inequalities for sums, series and integrals

Keywords:

generalized logarithmic mean

References:

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