Optimal bounds for Neuman-Sándor mean in terms of one-parameter centroidal mean
Abstract
In the paper, we find the best possible parameters $\alpha,\beta \in ({0,1})$ and $\lambda, \mu \in ({1/2,1})$ such that the double inequalities
$\sqrt{\alpha E^{2}( {a,b})+(1-\alpha)A^{2}({a,b})}<NS(a,b)<\sqrt{\beta E^{2}({a,b})+(1-\beta )A^{2}({a,b})}$,
$E[\lambda a+(1-\lambda)b,\lambda b+(1-\lambda)a]<NS(a,b)<E[\mu a+(1-\mu)b,\mu b+(1-\mu)a]$
holds for all $a,b>0$ with $a\ne b$, here $NS(a,b) = ({a-b}) / [2sin{h^{-1}}$ $(({a-b})/({a+b}))]$, $A(a,b)=({a+b})/2$ and $E(a,b)=2({a^{2}+ab+b^{2}})/[{3({a+b})}]$ are Neuman-S\'{a}ndor, arithmetic and centroidal means of two positive real numbers $a$ and $b$, respectively.- Published
- 10 May 2018
- Volume
- 2018 (2018)
- License
- CC BY 3.0
Copyright © 2018 Xiao Hong He, Hui Zuo Xu, Shao Yun Li, Su Qin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Cite this article
X. H. He, H. Z. Xu, S. Y. Li, S. Q. Wu, Optimal bounds for Neuman-Sándor mean in terms of one-parameter centroidal mean, Adv. Inequal. Appl. 2018 (2018), 8. https://doi.org/10.28919/aia/3599
BibTeX
@article{He2018,
author = {He, Xiao Hong and Xu, Hui Zuo and Li, Shao Yun and Wu, Su Qin},
title = {Optimal bounds for Neuman-Sándor mean in terms of one-parameter centroidal mean},
journal = {Adv. Inequal. Appl.},
volume = {2018},
year = {2018},
pages = {8},
doi = {10.28919/aia/3599}
}