## Abstract and Applied Analysis

### Bounds for the Combinations of Neuman-Sándor, Arithmetic, and Second Seiffert Means in terms of Contraharmonic Mean

#### Abstract

We give the greatest values ${r}_{\mathrm{1}}$, ${r}_{\mathrm{2}}$ and the least values ${s}_{\mathrm{1}}$, ${s}_{\mathrm{2}}$ in (1/2, 1) such that the double inequalities $C\left({r}_{\mathrm{1}}a+\left(\mathrm{1}-{r}_{\mathrm{1}}\right)b,{r}_{\mathrm{1}}b+\left(\mathrm{1}-{r}_{\mathrm{1}}\right)a\right)<\alpha A\left(a,b\right)+\left(\mathrm{1}-\alpha \right)T\left(a,b\right) and $C\left({r}_{\mathrm{2}}a+\left(\mathrm{1}-{r}_{\mathrm{2}}\right)b,{r}_{\mathrm{2}}b+\left(\mathrm{1}-{r}_{\mathrm{2}}\right)a\right)<\alpha A\left(a,b\right)+\left(\mathrm{1}-\alpha \right)M\left(a,b\right) hold for any $\alpha \in \left(\mathrm{0,1}\right)$ and all $a,b>\mathrm{0}$ with $a\ne b$, where $A\left(a,b\right)$, $M\left(a,b\right)$, $C\left(a,b\right),$ and $T\left(a,b\right)$ are the arithmetic, Neuman-Sándor, contraharmonic, and second Seiffert means of $a$ and $b$, respectively.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 903982, 5 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511799

Digital Object Identifier
doi:10.1155/2013/903982

Mathematical Reviews number (MathSciNet)
MR3035386

Zentralblatt MATH identifier
1272.26030

#### Citation

He, Zai-Yin; Qian, Wei-Mao; Jiang, Yun-Liang; Song, Ying-Qing; Chu, Yu-Ming. Bounds for the Combinations of Neuman-Sándor, Arithmetic, and Second Seiffert Means in terms of Contraharmonic Mean. Abstr. Appl. Anal. 2013 (2013), Article ID 903982, 5 pages. doi:10.1155/2013/903982. https://projecteuclid.org/euclid.aaa/1393511799

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