×

Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. (English) Zbl 1187.26015

Summary: For \(p\in\mathbb R\), the generalized logarithmic mean \(L_p(a,b)\), arithmetic mean \(A(a,b)\), and geometric mean \(G(a,b)\) of two positive numbers \(a\) and \(b\) are defined by \(L_p(a,b)=a\), for \(a=b\), \(L_p(a,b)= [(b^{p+1}-a^{p+1})/((p+1)(b-a))]^{1/p}\), for \(p\neq 0\), \(p\neq-1\), and \(a\neq b\), \(L_p(a,b)= (1/e)(b^b/a^a)^{1/(b-a)}\), for \(p=0\), and \(a\neq b\), \(L_p(a,b)= (b-a)/(\log b-\log a)\), for \(p=-1\), and \(a\neq b\), \(A(a,b)= (a+b)/2\), and \(G(a,b)= \sqrt{ab}\), respectively. In this paper, we find the greatest value \(p\) (or least value \(q\), resp.) such that the inequality \(L_p(a,b)<\alpha A(a,b)+ (1-\alpha) G(a,b)\) (or \(\alpha A(a,b)+ (1-\alpha)G(a,b)< L_q(a,b)\), resp.) holds for \(\alpha\in(0,1/2)\) (or \(\alpha\in(1/2,1)\), resp.) and all \(a,b>0\) with \(a\neq b\).

MSC:

26E60 Means
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Alzer H: Ungleichungen für Mittelwerte. Archiv der Mathematik 1986, 47(5):422-426. 10.1007/BF01189983 · Zbl 0585.26014 · doi:10.1007/BF01189983
[2] Alzer H, Qiu S-L: Inequalities for means in two variables. Archiv der Mathematik 2003, 80(2):201-215. 10.1007/s00013-003-0456-2 · Zbl 1020.26011 · doi:10.1007/s00013-003-0456-2
[3] Burk F: The geometric, logarithmic and arithmetic mean inequality. The American Mathematical Monthly 1987, 94(6):527-528. 10.2307/2322844 · Zbl 0632.26008 · doi:10.2307/2322844
[4] Janous W: A note on generalized Heronian means. Mathematical Inequalities & Applications 2001, 4(3):369-375. · Zbl 1128.26302 · doi:10.7153/mia-04-35
[5] Leach EB, Sholander MC: Extended mean values. II. Journal of Mathematical Analysis and Applications 1983, 92(1):207-223. 10.1016/0022-247X(83)90280-9 · Zbl 0517.26007 · doi:10.1016/0022-247X(83)90280-9
[6] Sándor J: On certain inequalities for means. Journal of Mathematical Analysis and Applications 1995, 189(2):602-606. 10.1006/jmaa.1995.1038 · Zbl 0822.26014 · doi:10.1006/jmaa.1995.1038
[7] Sándor J: On certain inequalities for means. II. Journal of Mathematical Analysis and Applications 1996, 199(2):629-635. 10.1006/jmaa.1996.0165 · Zbl 0854.26013 · doi:10.1006/jmaa.1996.0165
[8] Sándor J: On certain inequalities for means. III. Archiv der Mathematik 2001, 76(1):34-40. 10.1007/s000130050539 · Zbl 0976.26015 · doi:10.1007/s000130050539
[9] Shi, M-Y; Chu, Y-M; Jiang, Y-P, Optimal inequalities among various means of two arguments, No. 2009, 10 (2009) · Zbl 1187.26017
[10] Carlson BC: The logarithmic mean. The American Mathematical Monthly 1972, 79: 615-618. 10.2307/2317088 · Zbl 0241.33001 · doi:10.2307/2317088
[11] Sándor J: On the identric and logarithmic means. Aequationes Mathematicae 1990, 40(2-3):261-270. · Zbl 0717.26014 · doi:10.1007/BF02112299
[12] Sándor J: A note on some inequalities for means. Archiv der Mathematik 1991, 56(5):471-473. 10.1007/BF01200091 · Zbl 0693.26005 · doi:10.1007/BF01200091
[13] Lin TP: The power mean and the logarithmic mean. The American Mathematical Monthly 1974, 81: 879-883. 10.2307/2319447 · Zbl 0292.26015 · doi:10.2307/2319447
[14] Pittenger AO: Inequalities between arithmetic and logarithmic means. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika 1981, (678-715):15-18. · Zbl 0469.26009
[15] Imoru CO: The power mean and the logarithmic mean. International Journal of Mathematics and Mathematical Sciences 1982, 5(2):337-343. 10.1155/S0161171282000313 · Zbl 0483.26012 · doi:10.1155/S0161171282000313
[16] Chen Ch-P: The monotonicity of the ratio between generalized logarithmic means. Journal of Mathematical Analysis and Applications 2008, 345(1):86-89. 10.1016/j.jmaa.2008.03.071 · Zbl 1160.26012 · doi:10.1016/j.jmaa.2008.03.071
[17] Li X, Chen Ch-P, Qi F: Monotonicity result for generalized logarithmic means. Tamkang Journal of Mathematics 2007, 38(2):177-181. · Zbl 1132.26326
[18] Qi F, Chen Sh-X, Chen Ch-P: Monotonicity of ratio between the generalized logarithmic means. Mathematical Inequalities & Applications 2007, 10(3):559-564. · Zbl 1127.26021 · doi:10.7153/mia-10-52
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.