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**Sharp bounds by the generalized logarithmic mean for the geometric weighted mean of the geometric and harmonic means.**
*(English)*
Zbl 1244.26048

Summary: We present sharp upper and lower generalized logarithmic mean bounds for the geometric weighted mean of the geometric and harmonic means.

### MSC:

26D15 | Inequalities for sums, series and integrals |

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\textit{W.-M. Qian} and \textit{B.-Y. Long}, J. Appl. Math. 2012, Article ID 480689, 8 p. (2012; Zbl 1244.26048)

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### References:

[1] | K. B. Stolarsky, “The power and generalized logarithmic means,” The American Mathematical Monthly, vol. 87, no. 7, pp. 545-548, 1980. · Zbl 0455.26008 |

[2] | F. Qi and B.-N. Guo, “An inequality between ratio of the extended logarithmic means and ratio of the exponential means,” Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 229-237, 2003. · Zbl 1050.26020 |

[3] | C.-P. Chen and F. Qi, “Monotonicity properties for generalized logarithmic means,” The Australian Journal of Mathematical Analysis and Applications, vol. 1, no. 2, article 2, p. 4, 2004. · Zbl 1063.26025 |

[4] | X. Li, C.-P. Chen, and F. Qi, “Monotonicity result for generalized logarithmic means,” Tamkang Journal of Mathematics, vol. 38, no. 2, pp. 177-181, 2007. · Zbl 1132.26326 |

[5] | F. Qi, S.-X. Chen, and C.-P. Chen, “Monotonicity of ratio between the generalized logarithmic means,” Mathematical Inequalities & Applications, vol. 10, no. 3, pp. 559-564, 2007. · Zbl 1127.26021 |

[6] | C.-P. Chen, “The monotonicity of the ratio between generalized logarithmic means,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 86-89, 2008. · Zbl 1160.26012 |

[7] | B.-N. Guo and F. Qi, “A simple proof of logarithmic convexity of extended mean values,” Numerical Algorithms, vol. 52, no. 1, pp. 89-92, 2009. · Zbl 1184.26028 |

[8] | Y.-M. Chu and W.-F. Xia, “Inequalities for generalized logarithmic means,” Journal of Inequalities and Applications, vol. 2009, Article ID 763252, 7 pages, 2009. · Zbl 1187.26014 |

[9] | 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 |

[10] | W.-F. Xia and Y.-M. Chu, “Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means,” Revue d’Analyse Numérique et de Théorie de l’Approximation, vol. 39, no. 2, pp. 176-183, 2010. · Zbl 1249.26052 |

[11] | 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 |

[12] | 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 |

[13] | B.-Y. Long and Y.-M. Chu, “Optimal inequalities for generalized logarithmic, arithmetic, and geometric means,” Journal of Inequalities and Applications, vol. 2010, Article ID 806825, 10 pages, 2010. · Zbl 1187.26015 |

[14] | 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 |

[15] | M.-Y. Shi, Y.-M. Chu, and Y.-P. Jiang, “Optimal inequalities related to the power, harmonic and identric means,” Acta Mathematica Scientia A, vol. 31, no. 5, pp. 1377-1384, 2011 (Chinese). |

[16] | Y.-F. Qiu, M.-K. Wang, Y.-M. Chu, and G.-D. Wang, “Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean,” Journal of Mathematical Inequalities, vol. 5, no. 3, pp. 301-306, 2011. · Zbl 1226.26019 |

[17] | Y.-M. Chu, M.-K. Wang, and Z.-K. Wang, “A sharp double inequality between harmonic and identric means,” Abstract and Applied Analysis, vol. 2011, Article ID 657935, 7 pages, 2011. · Zbl 1225.26060 |

[18] | Y.-M. Chu and M.-K. Wang, “Optimal inequalities between harmonic, geometric, logarithmic, and arithmetic-geometric means,” Journal of Applied Mathematics, vol. 2011, Article ID 618929, 9 pages, 2011. · Zbl 1235.26010 |

[19] | Y.-M. Chu, S.-W. Hou, and W.-M. Gong, “Inequalities between logarithmic, harmonic, arithmetic and centroidal means,” SIAM Journal on Mathematical Analysis, vol. 2, no. 2, pp. 1-5, 2011. · Zbl 1312.26053 |

[20] | H.-N. Hu, S.-S. Wang, and Y.-M. Chu, “Optimal upper power mean bound for the convex combiantion of harmonic and logarithmic means,” Pacific Journal of Applied Mathematics, vol. 4, no. 1, pp. 35-44, 2011. |

[21] | Y.-F. Qiu, M.-K. Wang, and Y.-M. Chu, “The sharp combination bounds of arithmetic and logarithmic means for Seiffert’s mean,” International Journal of Pure and Applied Mathematics, vol. 72, no. 1, pp. 11-18, 2011. · Zbl 1244.26059 |

[22] | Y.-F. Qiu, M.-K. Wang, and Y.-M. Chu, “The optimal generalized Heronian mean bounds for the identric mean,” International Journal of Pure and Applied Mathematics, vol. 72, no. 1, pp. 19-26, 2011. · Zbl 1244.26060 |

[23] | M.-K. Wang, Z.-K. Wang, and Y.-M. Chu, “An optimal double inequality between geometric and identric means,” Applied Mathematics Letters, vol. 25, pp. 471-475, 2012. · Zbl 1247.26040 |

[24] | B.-N. Guo and F. Qi, “Inequalities for generalized weighted mean values of convex function,” Mathematical Inequalities & Applications, vol. 4, no. 2, pp. 195-202, 2001. · Zbl 0987.26019 |

[25] | A. O. Pittenger, “The logarithmic mean in n variables,” The American Mathematical Monthly, vol. 92, no. 2, pp. 99-104, 1985. · Zbl 0597.26027 |

[26] | P. Kahlig and J. Matkowski, “Functional equations involving the logarithmic mean,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 76, no. 7, pp. 385-390, 1996. · Zbl 0885.39008 |

[27] | G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, NJ, USA, 1951. · Zbl 0044.38301 |

[28] | E. B. Leach and M. C. Sholander, “Extended mean values. II,” Journal of Mathematical Analysis and Applications, vol. 92, no. 1, pp. 207-223, 1983. · Zbl 0517.26007 |

[29] | J. Sándor, “A note on some inequalities for means,” Archiv der Mathematik, vol. 56, no. 5, pp. 471-473, 1991. · Zbl 0693.26005 |

[30] | B. C. Carlson, “The logarithmic mean,” The American Mathematical Monthly, vol. 79, pp. 615-618, 1972. · Zbl 0241.33001 |

[31] | H. Alzer, “Ungleichungen für Mittelwerte,” Archiv der Mathematik, vol. 47, no. 5, pp. 422-426, 1986. · Zbl 0585.26014 |

[32] | P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalities, vol. 31, D. Reidel, Dordrecht, The Netherlands, 1988. · Zbl 0687.26005 |

[33] | H. Alzer, “Ungleichungen fur (e/a)a(b/e)b,” Elemente der Mathematik, vol. 40, no. 5, pp. 120-123, 1985. · Zbl 0579.20004 |

[34] | F. Burk, “Notes: the geometric, logarithmic, and arithmetic mean inequality,” The American Mathematical Monthly, vol. 94, no. 6, pp. 527-528, 1987. · Zbl 0632.26008 |

[35] | T. P. Lin, “The power mean and the logarithmic mean,” The American Mathematical Monthly, vol. 81, pp. 879-883, 1974. · Zbl 0292.26015 |

[36] | A. O. Pittenger, “Inequalities between arithmetic and logarithmic means,” Univerzitet u Beogradu, vol. 678-715, pp. 15-18, 1980. · Zbl 0469.26009 |

[37] | A. O. Pittenger, “The symmetric, logarithmic and power means,” Univerzitet u Beogradu, vol. 678-715, pp. 19-23, 1980. · Zbl 0469.26010 |

[38] | 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 |

[39] | Y.-M. Chu and B.-Y. Long, “Best possible inequalities between generalized logarithmic mean and classical means,” Abstract and Applied Analysis, vol. 2010, Article ID 303286, 13 pages, 2010. · Zbl 1185.26064 |

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