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Sharp power mean bounds for the combination of Seiffert and geometric means. (English) Zbl 1197.26054
Summary: We answer the question: for \(\alpha \in (0,1)\), what are the greatest value \(p\) and the least value \(q\) such that the double inequality \(M_{p}(a,b)<P^\alpha(a,b)G^{1-\alpha}(a,b)<M_q(a,b)\) holds for all \(a,b>0\) with \(a\neq b\). Here, \(M_{p}(a,b)\), \(P(a,b)\), and \(G(a,b)\) denote the power of order \(p\), Seiffert, and geometric means of two positive numbers \(a\) and \(b\), respectively.

MSC:
26E60 Means
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References:
[1] P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalities, vol. 31 of Mathematics and Its Applications, D. Reidel Publishing, Dordrecht, The Netherlands, 1988. · Zbl 0687.26005
[2] 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 · doi:10.1155/2010/604804 · eudml:226740
[3] M.-Y. Shi, Y.-M Chu, and Y.-P. Jiang, “Optimal inequalities among various means of two arguments,” Abstract and Applied Analysis, Article ID 694394, 10 pages, 2009. · Zbl 1187.26017 · doi:10.1155/2009/694394 · eudml:228533
[4] Y.-M Chu and W.-F. Xia, “Two sharp inequalities for power mean, geometric mean, and harmonic mean,” Journal of Inequalities and Applications, Article ID 741923, 6 pages, 2009. · Zbl 1187.26013 · doi:10.1155/2009/741923 · eudml:231537
[5] B.-Y. Long and Y.-M Chu, “Optimal power mean bounds for the weighted geometric mean of classical means,” Journal of Inequalities and Applications, Article ID 905679, 6 pages, 2010. · Zbl 1187.26016 · doi:10.1155/2010/905679 · eudml:227373
[6] 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
[7] 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 · doi:10.1007/s00013-003-0456-2
[8] F. Burk, “The geometric, logarithmic, and arithmetic mean inequality,” The American Mathematical Monthly, vol. 94, no. 6, pp. 527-528, 1987. · Zbl 0632.26008 · doi:10.2307/2322844
[9] H. Alzer and W. Janous, “Solution of problem 8\ast ,” Crux Mathematicorum with Mathematical Mayhem, vol. 13, pp. 173-178, 1987.
[10] H. Alzer, “Ungleichungen für Mittelwerte,” Archiv der Mathematik, vol. 47, no. 5, pp. 422-426, 1986. · Zbl 0585.26014 · doi:10.1007/BF01189983
[11] H. Alzer, “Ungleichungen für (e/a)a(b/e)b,” Elemente der Mathematik, vol. 40, pp. 120-123, 1985. · Zbl 0579.20004 · eudml:141366
[12] C. O. Imoru, “The power mean and the logarithmic mean,” International Journal of Mathematics and Mathematical Sciences, vol. 5, no. 2, pp. 337-343, 1982. · Zbl 0483.26012 · doi:10.1155/S0161171282000313 · eudml:44964
[13] 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, 1981. · Zbl 0469.26009
[14] 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, 1981. · Zbl 0469.26010
[15] K. B. Stolarsky, “The power and generalized logarithmic means,” The American Mathematical Monthly, vol. 87, no. 7, pp. 545-548, 1980. · Zbl 0455.26008 · doi:10.2307/2321420
[16] T. P. Lin, “The power mean and the logarithmic mean,” The American Mathematical Monthly, vol. 81, pp. 879-883, 1974. · Zbl 0292.26015 · doi:10.2307/2319447
[17] H.-J. Seiffert, “Problem 887,” Nieuw Archief voor Wiskunde, vol. 11, no. 2, pp. 176-176, 1993.
[18] E. Neuman and J. Sándor, “On the Schwab-Borchardt mean,” Mathematica Pannonica, vol. 14, no. 2, pp. 253-266, 2003. · Zbl 1053.26015 · eudml:228958
[19] A. A. Jagers, “Solution of problem 887,” Nieuw Archief voor Wiskunde, vol. 12, pp. 230-231, 1994.
[20] H.-J. Seiffert, “Ungleichungen für einen bestimmten Mittelwert,” Nieuw Archief voor Wiskunde. Vierde Serie, vol. 13, no. 2, pp. 195-198, 1995. · Zbl 0830.26008
[21] J. Sándor, “On certain inequalities for means. III,” Archiv der Mathematik, vol. 76, no. 1, pp. 34-40, 2001. · Zbl 0976.26015 · doi:10.1007/s000130050539
[22] M.-Y. Shi, Y.-M. Chu, and Y.-P. Jiang, “Three best inequalities for means in two variables,” International Mathematical Forum, vol. 5, no. 22, pp. 1059-1066, 2010. · Zbl 1206.26033 · www.m-hikari.com
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