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.