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Li, Yong-Min; Long, Bo-Yong; Chu, Yu-Ming; Gong, Wei-Ming

Optimal inequalities for power means. (English) Zbl 1244.26044

J. Appl. Math. 2012, Article ID 182905, 8 p. (2012).
Summary: We present the best possible power mean bounds for the product \(M^\alpha_p(a, b) M^{1-\alpha}_{-p}(a, b)\) for any \(p > 0, \alpha \in (0, 1)\), and all \(a, b > 0\) with \(a \neq b\). Here, \(M_p(a, b)\) is the \(p\)th power mean of two positive numbers \(a\) and \(b\).

Cited in 1 Document

MSC:

26D15 Inequalities for sums, series and integrals
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\textit{Y.-M. Li} et al., J. Appl. Math. 2012, Article ID 182905, 8 p. (2012; Zbl 1244.26044)
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References:

[1] Y.-M. Chu, S.-S. Wang, and C. Zong, “Optimal lower power mean bound for the convex combination of harmonic and logarithmic means,” Abstract and Applied Analysis, vol. 2011, Article ID 520648, 9 pages, 2011. · Zbl 1217.26040
[2] Y. Chu and B. Long, “Sharp inequalities between means,” Mathematical Inequalities & Applications, vol. 14, no. 3, pp. 647-655, 2011. · Zbl 1223.26049
[3] M.-K. Wang, Y.-M. Chu, Y.-F. Qiu, and S.-L. Qiu, “An optimal power mean inequality for the complete elliptic integrals,” Applied Mathematics Letters, vol. 24, no. 6, pp. 887-890, 2011. · Zbl 1216.33044
[4] B.-Y. Long and Y.-M. Chu, “Optimal power mean bounds for the weighted geometric mean of classical means,” Journal of Inequalities and Applications, vol. 2010, Article ID 905679, 6 pages, 2010. · Zbl 1187.26016
[5] W.-F. Xia, Y.-M. Chu, and G.-D. Wang, “The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means,” Abstract and Applied Analysis, vol. 2010, Article ID 604804, 9 pages, 2010. · Zbl 1190.26038
[6] Y.-M. Chu and W.-F. Xia, “Two optimal double inequalities between power mean and logarithmic mean,” Computers & Mathematics with Applications, vol. 60, no. 1, pp. 83-89, 2010. · Zbl 1205.26041
[7] Y.-M. Chu, Y.-F. Qiu, and M.-K. Wang, “Sharp power mean bounds for the combination of Seiffert and geometric means,” Abstract and Applied Analysis, vol. 2010, Article ID 108920, 12 pages, 2010. · Zbl 1197.26054
[8] M.-Y. Shi, Y.-M. Chu, and Y.-P. Jiang, “Optimal inequalities among various means of two arguments,” Abstract and Applied Analysis, vol. 2009, Article ID 694394, 10 pages, 2009. · Zbl 1187.26017
[9] Y.-M. Chu and W.-F. Xia, “Two sharp inequalities for power mean, geometric mean, and harmonic mean,” Journal of Inequalities and Applications, vol. 2009, Article ID 741923, 6 pages, 2009. · Zbl 1187.26013
[10] S. Wu, “Generalization and sharpness of the power means inequality and their applications,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 637-652, 2005. · Zbl 1083.26018
[11] P. A. Hästö, “Optimal inequalities between Seiffert’s mean and power means,” Mathematical Inequalities & Applications, vol. 7, no. 1, pp. 47-53, 2004. · Zbl 1049.26006
[12] H. Alzer and S.-L. Qiu, “Inequalities for means in two variables,” Archiv der Mathematik, vol. 80, no. 2, pp. 201-215, 2003. · Zbl 1020.26011
[13] H. Alzer, “A power mean inequality for the gamma function,” Monatshefte für Mathematik, vol. 131, no. 3, pp. 179-188, 2000. · Zbl 0964.33002
[14] J. E. Pe, “Generalization of the power means and their inequalities,” Journal of Mathematical Analysis and Applications, vol. 161, no. 2, pp. 395-404, 1991. · Zbl 0753.26009
[15] P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalities, vol. 31 of Mathematics and Its Applications (East European Series), D. Reidel Publishing Co., Dordrecht, The Netherlands, 1988. · Zbl 0687.26005
[16] S.-H. Wu and H.-N. Shi, “A relation of weak majorization and its applications to certain inequalities for means,” Mathematica Slovaca, vol. 61, no. 4, pp. 561-570, 2011. · Zbl 1265.26092
[17] S. Wu and L. Debnath, “Inequalities for differences of power means in two variables,” Analysis Mathematica, vol. 37, no. 2, pp. 151-159, 2011. · Zbl 1240.26069
[18] S. Wu, “On the weighted generalization of the Hermite-Hadamard inequality and its applications,” The Rocky Mountain Journal of Mathematics, vol. 39, no. 5, pp. 1741-1749, 2009. · Zbl 1181.26042
[19] H.-N. Shi, M. Bencze, S.-H. Wu, and D.-M. Li, “Schur convexity of generalized Heronian means involving two parameters,” Journal of Inequalities and Applications, vol. 2008, Article ID 879273, 9 pages, 2008. · Zbl 1165.26014
[20] S. Wu and L. Debnath, “Inequalities for convex sequences and their applications,” Computers & Mathematics with Applications, vol. 54, no. 4, pp. 525-534, 2007. · Zbl 1144.26016
[21] H.-N. Shi, S.-H. Wu, and F. Qi, “An alternative note on the Schur-convexity of the extended mean values,” Mathematical Inequalities & Applications, vol. 9, no. 2, pp. 219-224, 2006. · Zbl 1104.26014
[22] S. Wu, “Some results on extending and sharpening the Weierstrass product inequalities,” Journal of Mathematical Analysis and Applications, vol. 308, no. 2, pp. 689-702, 2005. · Zbl 1068.26024
[23] F. Burk, “The geomeric, logarithmic, and arithmetic mean inequality,” The American Mathematical Monthly, vol. 94, no. 6, pp. 527-528, 1987. · Zbl 0632.26008
[24] H. Alzer, “Ungleichungen für Mittelwerte,” Archiv der Mathematik, vol. 47, no. 5, pp. 422-426, 1986. · Zbl 0585.26014
[25] H. Alzer, “Ungleichungen für (e/a)a(b/e)b,” Elemente der Mathematik, vol. 40, pp. 120-123, 1985. · Zbl 0596.26014
[26] K. B. Stolarsky, “The power and generalized logarithmic means,” The American Mathematical Monthly, vol. 87, no. 7, pp. 545-548, 1980. · Zbl 0455.26008
[27] A. O. Pittenger, “Inequalities between arithmetic and logarithmic means,” Univerzitet u Beogradu. Publikacije Elektrotehni\vckog Fakulteta. Serija Matematika i Fizika, no. 678-715, pp. 15-18, 1980. · Zbl 0469.26009
[28] A. O. Pittenger, “The symmetric, logarithmic and power means,” Univerzitet u Beogradu. Publikacije Elektrotehni\vckog Fakulteta. Serija Matematika i Fizika, no. 678-715, pp. 19-23, 1980. · Zbl 0469.26010
[29] T. P. Lin, “The power mean and the logarithmic mean,” The American Mathematical Monthly, vol. 81, pp. 879-883, 1974. · Zbl 0292.26015
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