Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means
© B.-Y. Long and Y.-M. Chu. 2010
Received: 13 November 2009
Accepted: 25 February 2010
Published: 8 March 2010
For , the power mean of order of two positive numbers and is defined by , for , and , for . In this paper, we answer the question: what are the greatest value and the least value such that the double inequality holds for all and with ? Here , , and denote the classical arithmetic, geometric, and harmonic means, respectively.
Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities for can be found in literatures [1–12]. It is well known that is continuous and increasing with respect to for fixed and . Let , and be the classical arithmetic, geometric, and harmonic means of two positive numbers and , respectively. Then
In , Mao proved
The following sharp bounds for and in terms of power mean are proved in :
2. Main Result
In order to establish our main results we need the following lemma.
Therefore, Lemma 2.1(2) follows from (2.1)–(2.3) and (2.5) together with (2.8).
This work is partly supported by the National Natural Science Foundation of China (Grant no. 60850005) and the Natural Science Foundation of Zhejiang Province (Grant no. D7080080, Y607128).
- Wu S: Generalization and sharpness of the power means inequality and their applications. Journal of Mathematical Analysis and Applications 2005, 312(2):637–652. 10.1016/j.jmaa.2005.03.050MathSciNetView ArticleMATHGoogle Scholar
- Richards KC: Sharp power mean bounds for the Gaussian hypergeometric function. Journal of Mathematical Analysis and Applications 2005, 308(1):303–313. 10.1016/j.jmaa.2005.01.018MathSciNetView ArticleMATHGoogle Scholar
- Wang WL, Wen JJ, Shi HN: Optimal inequalities involving power means. Acta Mathematica Sinica 2004, 47(6):1053–1062.MathSciNetMATHGoogle Scholar
- Hästö PA: Optimal inequalities between Seiffert's mean and power means. Mathematical Inequalities & Applications 2004, 7(1):47–53.MathSciNetView ArticleMATHGoogle Scholar
- Alzer H: A power mean inequality for the gamma function. Monatshefte für Mathematik 2000, 131(3):179–188. 10.1007/s006050070007MathSciNetView ArticleMATHGoogle Scholar
- Alzer H, Qiu S-L: Inequalities for means in two variables. Archiv der Mathematik 2003, 80(2):201–215. 10.1007/s00013-003-0456-2MathSciNetView ArticleMATHGoogle Scholar
- Tarnavas CD, Tarnavas DD: An inequality for mixed power means. Mathematical Inequalities & Applications 1999, 2(2):175–181.MathSciNetView ArticleMATHGoogle Scholar
- Bukor J, Tóth J, Zsilinszky L: The logarithmic mean and the power mean of positive numbers. Octogon Mathematical Magazine 1994, 2(1):19–24.MathSciNetGoogle Scholar
- Pečarić JE: Generalization of the power means and their inequalities. Journal of Mathematical Analysis and Applications 1991, 161(2):395–404. 10.1016/0022-247X(91)90339-2MathSciNetView ArticleMATHGoogle Scholar
- Chen J, Hu B: The identric mean and the power mean inequalities of Ky Fan type. Facta Universitatis. Series: Mathematics and Informatics 1989, (4):15–18.MATHGoogle Scholar
- Imoru CO: The power mean and the logarithmic mean. International Journal of Mathematics and Mathematical Sciences 1982, 5(2):337–343. 10.1155/S0161171282000313MathSciNetView ArticleMATHGoogle Scholar
- Lin TP: The power mean and the logarithmic mean. The American Mathematical Monthly 1974, 81: 879–883. 10.2307/2319447MathSciNetView ArticleMATHGoogle Scholar
- Alzer H, Janous W: Solution of problem . Crux Mathematicorum 1987, 13: 173–178.Google Scholar
- Bullen PS, Mitrinović DS, Vasić PM: Means and Their Inequalities, Mathematics and Its Applications. Volume 31. D. Reidel, Dordrecht, The Netherlands; 1988:xx+459.View ArticleGoogle Scholar
- Mao Q-J: Power mean, logarithmic mean and Heronian dual mean of two positive number. Journal of Suzhou College of Education 1999, 16(1–2):82–85.Google Scholar
- Chu Y-M, Xia W-F: Two sharp inequalities for power mean, geometric mean, and harmonic mean. Journal of Inequalities and Applications 2009, 2009:-6.Google Scholar
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