## Abstract and Applied Analysis

### Sharp Power Mean Bounds for Sándor Mean

#### Abstract

We prove that the double inequality ${M}_{p}(a,b) holds for all $a,b>0$ with $a\ne b$ if and only if $p\le 1/3$ and $q\ge \text{l}\text{o}\text{g} 2/(1+\text{l}\text{o}\text{g} 2)=0.4093$…, where $X(a,b)$ and ${M}_{r}(a,b)$ are the Sándor and $r$th power means of $a$ and $b$, respectively.

#### Article information

Source
Abstr. Appl. Anal., Volume 2015 (2015), Article ID 172867, 5 pages.

Dates
First available in Project Euclid: 15 April 2015

https://projecteuclid.org/euclid.aaa/1429103741

Digital Object Identifier
doi:10.1155/2015/172867

Mathematical Reviews number (MathSciNet)
MR3303262

Zentralblatt MATH identifier
1372.26032

#### Citation

Chu, Yu-Ming; Yang, Zhen-Hang; Wu, Li-Min. Sharp Power Mean Bounds for Sándor Mean. Abstr. Appl. Anal. 2015 (2015), Article ID 172867, 5 pages. doi:10.1155/2015/172867. https://projecteuclid.org/euclid.aaa/1429103741

#### References

• P. S. Bullen, D. S. Mitrinovic, and P. M. Vasić, Means and Their Inequalities, D. Reidel, Dordrecht, The Netherlands, 1988.
• H.-J. Seiffert, “Aufgabe $\beta$16,” Die Wurzel, vol. 29, pp. 221–222, 1995.
• A. A. Jagers, “Solution of problem 887,” Nieuw Archief voor Wiskunde, vol. 12, no. 4, pp. 230–231, 1994.
• P. A. Hästö, “A monotonicity property of ratios of symmetric homogeneous means,” Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 5, article 71, 23 pages, 2002.
• P. A. Hästö, “Optimal inequalities between Seiffert's mean and power means,” Mathematical Inequalities & Applications, vol. 7, no. 1, pp. 47–53, 2004.
• A. Witkowski, “Interpolations of Schwab-Borchardt mean,” Mathematical Inequalities & Applications, vol. 16, no. 1, pp. 193–206, 2013.
• I. Costin and G. Toader, “Optimal evaluations of some Seiffert-type means by power means,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4745–4754, 2013.
• Y.-M. Chu and B.-Y. Long, “Bounds of the Neuman-Sándor mean using power and identric means,” Abstract and Applied Analysis, vol. 2013, Article ID 832591, 6 pages, 2013.
• H. Alzer, “Ungleichungen für mittelwerte,” Archiv der Mathematik, vol. 47, no. 5, pp. 422–426, 1986.
• H. Alzer, “Ungleichungen für (e/a)$^{a}$(b/e)$^{b}$,” Elementary Mathematics, vol. 40, pp. 120–123, 1985.
• F. Burk, “The geometric, logarithmic, and arithmetic mean inequality,” The American Mathematical Monthly, vol. 94, no. 6, pp. 527–528, 1987.
• T. P. Lin, “The power mean and the logarithmic mean,” The American Mathematical Monthly, vol. 81, pp. 879–883, 1974.
• A. O. Pittenger, “Inequalities between arithmetic and logarithmic means,” Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, no. 678–715, pp. 15–18, 1980.
• A. O. Pittenger, “The symmetric, logarithmic and power means,” Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, no. 678–715, pp. 19–23, 1980.
• K. B. Stolarsky, “The power and generalized logarithmic means,” The American Mathematical Monthly, vol. 87, no. 7, pp. 545–548, 1980.
• H. Alzer and S.-L. Qiu, “Inequalities for means in two variables,” Archiv der Mathematik, vol. 80, no. 2, pp. 201–215, 2003.
• J. Sándor, “Two sharp inequalities for trigonometric and hyperbolic functions,” Mathematical Inequalities &Applications, vol. 15, no. 2, pp. 409–413, 2012.
• J. Sándor, “On two new means of two variables,” Notes on Number Theory and Discrete Mathematics, vol. 20, no. 1, pp. 1–9, 2014.
• J. Sándor, “On certain inequalities for means III,” Archiv der Mathematik, vol. 76, no. 1, pp. 34–40, 2001. \endinput