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Sharp Bounds for Neuman Means in Terms of Geometric, Arithemtic and Quadratic Means

In this paper, we find the greatest values $α_{1}$, $α_{2}$, $α_{3}$, $α_{4}$, $α_{5}$, $α_{6}$, $α_{7}$, $α_{8}$ and the least values $β_{1}$, $β_{2}$, $β_{3}$, $β_{4}$, $β_{5}$, $β_{6}$, $β_{7}$, $β_{8}$ such that the double inequalities $$A^{α_{1}}(a,b)G^{1-α_{1}}(a,b)<N_{GA}(a,b)<A^{β_{1}}(a,b)G^{1-β_{1}}(a,b),$$ $$\frac{α_{2}}{G(a,b)}+\frac{1-α_{2}}{A(a,b)}<\frac{1}{N_{GA}(a,b)}<\frac{β_{2}}{G(a,b)}+\frac{1-β_{2}}{A(a,b)},$$ $$A^{α_{3}}(a,b)G^{1-α_{3}}(a,b)<N_{AG}(a,b)<A^{β_{3}}(a,b)G^{1-β_{3}}(a,b),$$ $$\frac{α_{4}}{G(a,b)}+\frac{1-α_{4}}{A(a,b)}<\frac{1}{N_{AG}(a,b)}<\frac{β_{4}}{G(a,b)}+\frac{1-β_{4}}{A(a,b)},$$ $$Q^{α_{5}}(a,b)A^{1-α_{5}}(a,b)<N_{AQ}(a,b)<Q^{β_{5}}(a,b)A^{1-β_{5}}(a,b),$$ $$\frac{α_{6}}{A(a,b)}+\frac{1-α_{6}}{Q(a,b)}<\frac{1}{N_{AQ}(a,b)}<\frac{β_{6}}{A(a,b)}+\frac{1-β_{6}}{Q(a,b)},$$ $$Q^{α_{7}}(a,b)A^{1-α_{7}}(a,b)<N_{QA}(a,b)<Q^{β_{7}}(a,b)A^{1-β_{7}}(a,b),$$ $$\frac{α_{8}}{A(a,b)}+\frac{1-α_{8}}{Q(a,b)}<\frac{1}{N_{QA}(a,b)}<\frac{β_{8}}{A(a,b)}+\frac{1-β_{8}}{Q(a,b)}$$ 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.