## Abstract

For $a,b>\mathrm{0}$ with $a\ne b$, the Schwab-Borchardt mean $\text{S}\text{B}(a,b)$ is defined as $$. In this paper, we find the greatest values of ${\alpha}_{\mathrm{1}}$ and ${\alpha}_{\mathrm{2}}$ and the least values of ${\beta}_{\mathrm{1}}$ and ${\beta}_{\mathrm{2}}$ in $[\mathrm{0,1}/\mathrm{2}]$ such that $$ and $$. Similarly, we also find the greatest values of ${\alpha}_{\mathrm{3}}$ and ${\alpha}_{\mathrm{4}}$ and the least values of ${\beta}_{\mathrm{3}}$ and ${\beta}_{\mathrm{4}}$ in $[\mathrm{1}/\mathrm{2,1}]$ such that $$ and $$. Here, $H(a,b)=\mathrm{2}ab/(a+b)$, $A(a,b)=(a+b)/\mathrm{2}$, and $C(a,b)=({a}^{\mathrm{2}}+{b}^{\mathrm{2}})/(a+b)$ are the harmonic, arithmetic, and contraharmonic means, respectively, and ${S}_{HA}(a,b)=\text{S}\text{B}(H,A)$, ${S}_{AH}(a,b)=\text{S}\text{B}(A,H)$, ${S}_{CA}(a,b)=\text{S}\text{B}(C,A)$, and ${S}_{AC}(a,b)=\text{S}\text{B}(A,C)$ are four Neuman means derived from the Schwab-Borchardt mean.

## Citation

Zai-Yin He. Yu-Ming Chu. Miao-Kun Wang. "Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means." J. Appl. Math. 2013 1 - 4, 2013. https://doi.org/10.1155/2013/807623