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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:
 [1] Alzer, H., On an inequality of H. minc and L. sathre, J. math. anal. appl., 179, 396-402, (1993) · Zbl 0792.26008 [2] Chen, Ch.-P.; Qi, F., Extension of an inequality of H. Alzer for negative powers, Tamkang J. math., 36, 1, 69-72, (2005) · Zbl 1068.26012 [3] Chen, Ch.-P.; Qi, F., Extension of an inequality of H. Alzer, Math. gaz., 90, 293-294, (2006) [4] Chen, Ch.-P.; Qi, F.; Cerone, P.; Dragomir, S.S., Monotonicity of sequences involving convex and concave functions, Math. inequal. appl., 6, 2, 229-239, (2003) · Zbl 1032.26021 [5] Chen, Ch.-P.; Qi, F., Generalization of an inequality of Alzer for negative powers, Tamkang J. math., 36, 3, 219-222, (2005) · Zbl 1079.26014 [6] Chen, Ch.-P.; Qi, F., Note on Alzer’s inequality, Tamkang J. math., 37, 1, 11-14, (2006) · Zbl 1116.26015 [7] Elezović, N.; Pečarić, J., On Alzer’s inequality, J. math. anal. appl., 223, 1, 366-369, (1998) · Zbl 0911.26013 [8] Minc, H.; Sathre, L., Some inequalities involving $$(r!)^{1 / r}$$, Proc. edinb. math. soc., 14, 41-46, (1964/1965) · Zbl 0124.01003 [9] Qi, F., Generalization of H. Alzer’s inequality, J. math. anal. appl., 240, 294-297, (1999) · Zbl 0946.26007 [10] Qi, F.; Guo, B.-N., Monotonicity of sequences involving convex function and sequence, Math. inequal. appl., 9, 2, 247-254, (2006) · Zbl 1093.26024 [11] Sándor, J., On an inequality of Alzer, J. math. anal. appl., 192, 1034-1035, (1995) · Zbl 0829.26013 [12] Stolarsky, K.B., Generalizations of the logarithmic Mean, Math. mag., 48, 87-92, (1975) · Zbl 0302.26003 [13] Stolarsky, K.B., The power and generalized logarithmic means, Amer. math. monthly, 87, 545-548, (1980) · Zbl 0455.26008 [14] Ume, J.S., An elementary proof of H. Alzer’s inequality, Math. jpn., 44, 3, 521-522, (1996) · Zbl 0865.26016 [15] Xu, Z.-K.; Xu, D.-P., A general form of Alzer’s inequality, Comput. math. appl., 44, 365-373, (2002) · Zbl 1056.26016
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