Sharp bounds for the arithmetic-geometric mean. (English) Zbl 1308.26058
Summary: In this article, we establish some new inequality chains for the ratio of certain bivariate means, and we present several sharp bounds for the arithmetic-geometric mean.
MSC:
| 26E60 | Means |
| 26D07 | Inequalities involving other types of functions |
| 33E05 | Elliptic functions and integrals |
Keywords:
arithmetic-geometric mean; Stolarsky mean; Gini mean; complete elliptic integral of the first kindReferences:
| [1] | Borwein JM, Borwein PB: Pi and the AGM. Wiley, New York; 1987. · Zbl 0611.10001 |
| [2] | Stolarsky KB: Generalizations of the logarithmic mean.Math. Mag. 1975, 48:87-92. 10.2307/2689825 · Zbl 0302.26003 · doi:10.2307/2689825 |
| [3] | Gini C: Di una formula comprensiva delle medie.Metron 1938,13(2):3-22. · Zbl 0018.41404 |
| [4] | Carlson BC, Vuorinen M: Inequality of the AGM and the logarithmic mean.SIAM Rev. 1991,33(4):653-654. · doi:10.1137/1033141 |
| [5] | Sándor J: On certain inequalities for means.J. Math. Anal. Appl. 1995,189(2):602-606. 10.1006/jmaa.1995.1038 · Zbl 0822.26014 · doi:10.1006/jmaa.1995.1038 |
| [6] | Qi F, Sofo A: An alternative and united proof of a double inequality for bounding the arithmetic-geometric mean.Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 2009,71(3):69-76. · Zbl 1299.26068 |
| [7] | Neuman E, Sándor J: On certain means of two arguments and their extensions.Int. J. Math. Math. Sci. 2003, 16:981-993. · Zbl 1040.26015 · doi:10.1155/S0161171203208103 |
| [8] | Yang Z-H: A new proof of inequalities for Gauss compound mean.Int. J. Math. Anal. 2010,4(21-24):1013-1018. · Zbl 1206.26023 |
| [9] | Vamanamurthy MK, Vuorinen M: Inequalities for means.J. Math. Anal. Appl. 1994,183(1):155-166. 10.1006/jmaa.1994.1137 · Zbl 0802.26009 · doi:10.1006/jmaa.1994.1137 |
| [10] | Borwein JM, Borwein PB: Inequalities for compound mean iterations with logarithmic asymptotes.J. Math. Anal. Appl. 1993,177(2):572-582. 10.1006/jmaa.1993.1278 · Zbl 0783.33001 · doi:10.1006/jmaa.1993.1278 |
| [11] | Alzer H, Qiu S-L: Monotonicity theorems and inequalities for the complete elliptic integrals.J. Comput. Appl. Math. 2004,172(2):289-312. 10.1016/j.cam.2004.02.009 · Zbl 1059.33029 · doi:10.1016/j.cam.2004.02.009 |
| [12] | Qiu S-L, Shen J-M: On two problems concerning means.J. Hangzhou Inst. Electron. Eng. 1997,17(3):1-7. (in Chinese) |
| [13] | Sándor J: On certain inequalities for means II.J. Math. Anal. Appl. 1996,199(2):629-635. 10.1006/jmaa.1996.0165 · Zbl 0854.26013 · doi:10.1006/jmaa.1996.0165 |
| [14] | Sándor J: On certain inequalities for means III.Arch. Math. 2001,76(1):34-40. 10.1007/s000130050539 · Zbl 0976.26015 · doi:10.1007/s000130050539 |
| [15] | Toader G: Some mean values related to the arithmetic-geometric mean.J. Math. Anal. Appl. 1998,218(2):358-368. 10.1006/jmaa.1997.5766 · Zbl 0892.26015 · doi:10.1006/jmaa.1997.5766 |
| [16] | Chung S-Y: Functional means and harmonic functional means.Bull. Aust. Math. Soc. 1998,57(2):207-220. 10.1017/S0004972700031609 · Zbl 0909.26010 · doi:10.1017/S0004972700031609 |
| [17] | Bracken P: An arithmetic-geometric mean inequality.Expo. Math. 2001,19(3):273-279. 10.1016/S0723-0869(01)80006-2 · Zbl 1135.26302 · doi:10.1016/S0723-0869(01)80006-2 |
| [18] | Liu Z: Compounding of Stolarsky means.Soochow J. Math. 2004,30(2):149-163. · Zbl 1062.26026 |
| [19] | Liu Z: Remarks on arithmetic-geometric mean and geometric-harmonic mean.J. Anshan Univ. Sci. Technol. 2007,30(3):230-235. (in Chinese) |
| [20] | Neuman E: Inequalities for weighted sums of powers and their applications.Math. Inequal. Appl. 2012,15(4):995-1005. · Zbl 1253.26041 |
| [21] | Yang Z-H: The log-convexity of another class of one-parameter means and its applications.Bull. Korean Math. Soc. 2012,49(1):33-47. 10.4134/BKMS.2012.49.1.033 · Zbl 1238.26031 · doi:10.4134/BKMS.2012.49.1.033 |
| [22] | Losonczi L: Ratio of Stolarsky means: monotonicity and comparison.Publ. Math. (Debr.) 2009,75(1-2):221-238. · Zbl 1212.26074 |
| [23] | Yang, Z-H, Some monotonicity results for the ratio of two-parameter symmetric homogeneous functions, No. 2009 (2009) · Zbl 1186.26027 |
| [24] | Yang Z-H: Log-convexity of ratio of the two-parameter symmetric homogeneous functions and an application.J. Inequal. Spec. Funct. 2010,1(1):16-29. · Zbl 1312.26028 |
| [25] | Yang, Z-H, The monotonicity results for the ratio of certain mixed means and their applications, No. 2012 (2012) · Zbl 1260.26037 |
| [26] | Peetre J: Generalizing the arithmetic geometric mean - a hapless computer experiment.Int. J. Math. Math. Sci. 1989,12(2):235-245. 10.1155/S016117128900027X · Zbl 0707.26005 · doi:10.1155/S016117128900027X |
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