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) 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.
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