## Proof of one optimal inequality for generalized logarithmic, arithmetic, and geometric means.(English)Zbl 1210.26033

The main theorem is an answer on two open problems given by B.-Y. Long and Y.-M. Chu [J. Inequal. Appl. 2010, Article ID 806825 (2010; Zbl 1187.26015)].
Theorem: Let $$\alpha \in (0,1/2)\cup (1/2,1)$$, $$a\not= b$$, $$a>0$$, $$b>0$$. Let $$p(\alpha)$$ be a solution of $\frac 1p \ln (1+p)+\ln (\frac{\alpha}{2})=0$ in $$(-1,1)$$. Then, if $$\alpha \in (0,1/2)$$, then $\alpha A(a,b)+(1-\alpha)G(a,b) <L_p(a,b)$ for $$p\geq p(\alpha)$$ and $$p(\alpha)$$ is the best constant, and if $$\alpha \in (1/2,1)$$, then $\alpha A(a,b)+(1-\alpha)G(a,b) >L_p(a,b)$ for $$p\leq p(\alpha)$$ and $$p(\alpha)$$ is the best constant.
$$A(a,b), G(a,b)$$ and $$L_p(a,b)$$ are notations for an arithmetic mean, a geometric mean and a generalized logarithmic mean respectively.

### MSC:

 26E60 Means 26D99 Inequalities in real analysis

Zbl 1187.26015
Full Text:

### References:

  Long B-Y, Chu Y-M: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. Journal of Inequalities and Applications 2010, 2010:-10. · Zbl 1187.26015  Alzer H: Ungleichungen für Mittelwerte. Archiv der Mathematik 1986, 47(5):422-426. 10.1007/BF01189983 · Zbl 0585.26014  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  Burk F: The geometric, logarithmic, and arithmetic mean inequality. The American Mathematical Monthly 1987, 94(6):527-528. 10.2307/2322844 · Zbl 0632.26008  Janous W: A note on generalized Heronian means. Mathematical Inequalities & Applications 2001, 4(3):369-375. · Zbl 1128.26302  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  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  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  Sándor J: On certain inequalities for means. III. Archiv der Mathematik 2001, 76(1):34-40. 10.1007/s000130050539 · Zbl 0976.26015  Shi M-Y, Chu Y-M, Jiang Y-P: Optimal inequalities among various means of two arguments. Abstract and Applied Analysis 2009, 2009:-10. · Zbl 1187.26017  Carlson BC: The logarithmic mean. The American Mathematical Monthly 1972, 79: 615-618. 10.2307/2317088 · Zbl 0241.33001  Sándor J: On the identric and logarithmic means. Aequationes Mathematicae 1990, 40(2-3):261-270. · Zbl 0717.26014  Sándor J: A note on some inequalities for means. Archiv der Mathematik 1991, 56(5):471-473. 10.1007/BF01200091 · Zbl 0693.26005  Lin TP: The power mean and the logarithmic mean. The American Mathematical Monthly 1974, 81: 879-883. 10.2307/2319447 · Zbl 0292.26015  Pittenger AO: Inequalities between arithmetic and logarithmic means. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta 1980, (678-715):15-18. · Zbl 0469.26009  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  Chen C-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  Li X, Chen C-P, Qi F: Monotonicity result for generalized logarithmic means. Tamkang Journal of Mathematics 2007, 38(2):177-181. · Zbl 1132.26326  Qi F, Chen S-X, Chen C-P: Monotonicity of ratio between the generalized logarithmic means. Mathematical Inequalities & Applications 2007, 10(3):559-564. · Zbl 1127.26021
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