Bounds for the Arithmetic Mean in Terms of the Neuman-Sándor and Other Bivariate Means

Abstract

We present the largest values ${\alpha }_{\mathrm{1}}$, ${\alpha }_{\mathrm{2}}$, and ${\alpha }_{\mathrm{3}}$ and the smallest values ${\beta }_{\mathrm{1}}$, ${\beta }_{\mathrm{2}}$, and ${\beta }_{\mathrm{3}}$ such that the double inequalities  $(\mathrm{1}-{\beta }_{\mathrm{1}})H(a,b)$, ${\alpha }_{\mathrm{2}}M(a,b)+(\mathrm{1}-{\alpha }_{\mathrm{2}})$ $\stackrel{-}{H}(a,b)$  , and   $(a,b)+(\mathrm{1}-{\beta }_{\mathrm{3}})He(a,b)$ hold for all $a,b>\mathrm{0}$ with $a\ne b$, where $M(a,b)$, $A(a,b)$, $He(a,b)$, $H(a,b)$ and $\stackrel{-}{H}(a,b)$ denote the Neuman-Sándor, arithmetic, Heronian, harmonic, and harmonic root-square means of $a$ and $b$, respectively.