## Journal of Applied Mathematics

### Bounds for the Arithmetic Mean in Terms of the Neuman-Sándor and Other Bivariate Means

#### Abstract

We present the largest values ${\alpha }_{\mathrm{1}}$, ${\alpha }_{\mathrm{2}}$, and ${\alpha }_{\mathrm{3}}$ and the smallest values ${\beta }_{\mathrm{1}}$, ${\beta }_{\mathrm{2}}$, and ${\beta }_{\mathrm{3}}$ such that the double inequalities ${\alpha }_{\mathrm{1}}M\left(a,b\right)+\left(\mathrm{1}-{\alpha }_{\mathrm{1}}\right)H\left(a,b\right)$\left(\mathrm{1}-{\beta }_{\mathrm{1}}\right)H\left(a,b\right)$, ${\alpha }_{\mathrm{2}}M\left(a,b\right)+\left(\mathrm{1}-{\alpha }_{\mathrm{2}}\right)$$\stackrel{-}{H}\left(a,b\right)$$<$$A\left(a,b\right)<{\beta }_{\mathrm{2}}M\left(a,b\right)+\left(\mathrm{1}-{\beta }_{\mathrm{2}}\right)\stackrel{-}{H}\left(a,b\right)$, and ${\alpha }_{\mathrm{3}}M\left(a,b\right)+\left(\mathrm{1}-{\alpha }_{\mathrm{3}}\right)He\left(a,b\right)<$$A\left(a,b\right)<{\beta }_{\mathrm{3}}M$$\left(a,b\right)+\left(\mathrm{1}-{\beta }_{\mathrm{3}}\right)He\left(a,b\right)$ hold for all $a,b>\mathrm{0}$ with $a\ne b$, where $M\left(a,b\right)$, $A\left(a,b\right)$, $He\left(a,b\right)$, $H\left(a,b\right)$ and $\stackrel{-}{H}\left(a,b\right)$ denote the Neuman-Sándor, arithmetic, Heronian, harmonic, and harmonic root-square means of $a$ and $b$, respectively.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 582504, 7 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394808318

Digital Object Identifier
doi:10.1155/2013/582504

Mathematical Reviews number (MathSciNet)
MR3147875

Zentralblatt MATH identifier
06950758

#### Citation

Zhang, Fan; Chu, Yu-Ming; Qian, Wei-Mao. Bounds for the Arithmetic Mean in Terms of the Neuman-Sándor and Other Bivariate Means. J. Appl. Math. 2013 (2013), Article ID 582504, 7 pages. doi:10.1155/2013/582504. https://projecteuclid.org/euclid.jam/1394808318

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