Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means

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.