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

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.

### MSC:

 26D15 Inequalities for sums, series and integrals

### Keywords:

generalized logarithmic mean
Full Text:

### References:

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