### 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:**2018-05-10

**How to Cite this Article:**Xiao Hong He, Hui Zuo Xu, Shao Yun Li, Su Qin Wu, Optimal bounds for Neuman-Sándor mean in terms of one-parameter centroidal mean, Advances in Inequalities and Applications, Vol 2018 (2018), Article ID 8 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.

Advances in Inequalities and Applications

ISSN 2050-7461

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