×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.