In this paper, we find the greatest values $\alpha_{ 1}$, $\alpha_{ 2}$, $\alpha_{ 3}$, $\alpha_{ 4}$, $\alpha_{ 5}$, $\alpha_{ 6}$, $\alpha_{ 7}$, $\alpha_{ 8}$ and the least values $\beta_{ 1}$, $\beta_{ 2}$, $\beta_{ 3}$, $\beta_{ 4}$, $\beta_{ 5}$, $\beta_{ 6}$, $\beta_{ 7}$, $\beta_{ 8}$ such that the double inequalities $$A^{ \alpha_{ 1}}(a,b)G^{ 1-\alpha_{ 1}}(a,b)<N_{ GA}(a,b)<A^{ \beta_{ 1}}(a,b)G^{ 1-\beta_{ 1}}(a,b),$$ $$\frac{ \alpha_{ 2}}{ G(a,b)}+\frac{ 1-\alpha_{ 2}}{ A(a,b)}<\frac{ 1}{ N_{ GA}(a,b)}<\frac{ \beta_{ 2}}{ G(a,b)}+\frac{ 1-\beta_{ 2}}{ A(a,b)},$$ $$A^{ \alpha_{ 3}}(a,b)G^{ 1-\alpha_{ 3}}(a,b)<N_{ AG}(a,b)<A^{ \beta_{ 3}}(a,b)G^{ 1-\beta_{ 3}}(a,b),$$ $$\frac{ \alpha_{ 4}}{ G(a,b)}+\frac{ 1-\alpha_{ 4}}{ A(a,b)}<\frac{ 1}{ N_{ AG}(a,b)}<\frac{ \beta_{ 4}}{ G(a,b)}+\frac{ 1-\beta_{ 4}}{ A(a,b)},$$ $$Q^{ \alpha_{ 5}}(a,b)A^{ 1-\alpha_{ 5}}(a,b)<N_{ AQ}(a,b)<Q^{ \beta_{ 5}}(a,b)A^{ 1-\beta_{ 5}}(a,b),$$ $$\frac{ \alpha_{ 6}}{ A(a,b)}+\frac{ 1-\alpha_{ 6}}{ Q(a,b)}<\frac{ 1}{ N_{ AQ}(a,b)}<\frac{ \beta_{ 6}}{ A(a,b)}+\frac{ 1-\beta_{ 6}}{ Q(a,b)},$$ $$Q^{ \alpha_{ 7}}(a,b)A^{ 1-\alpha_{ 7}}(a,b)<N_{ QA}(a,b)<Q^{ \beta_{ 7}}(a,b)A^{ 1-\beta_{ 7}}(a,b),$$ $$\frac{ \alpha_{ 8}}{ A(a,b)}+\frac{ 1-\alpha_{ 8}}{ Q(a,b)}<\frac{ 1}{ N_{ QA}(a,b)}<\frac{ \beta_{ 8}}{ A(a,b)}+\frac{ 1-\beta_{ 8}}{ Q(a,b)}$$ hold for all $a, b>0$ with $a\neq b$, where $G$, $A$ and $Q$ are respectively the geometric, arithmetic and quadratic means, and $N_{ GA}$, $N_{ AG}$, $N_{ AQ}$ and $N_{ QA}$ are the Neuman means derived from the Schwab-Borchardt mean.
Categories: math.CA math.OC